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Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Fractional Edge Coloring
Debbie Berg
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Material from
1 Fractional Graph Theory: A Rational Approach to the Theory ofGraphs, by Edward Scheinerman and Daniel Ullman
2 Stephen Hartke
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Definitions (Review)
� A hypergraph H is a pair (S ,X ) where S is a finite set and Xconsists of subsets of S , called hyperedges.
� Note that a graph is a hypergraph where we restrict the subsetsto size 2.
� A covering of H is a collection of hyperedges such that everyelement of S is contained in some hyperedge.
� The covering number k(H) of a hypergraph H is the number ofhyperedges in the smallest covering of H.
� A t-fold covering of H is a collection of hyperedges such thatevery element of S is contained in at least t hyperedges.
� The t-fold covering number kt(H) is the number of hyperedgesin the smallest t-fold covering of H.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Definitions (Review)
� A hypergraph H is a pair (S ,X ) where S is a finite set and Xconsists of subsets of S , called hyperedges.
� Note that a graph is a hypergraph where we restrict the subsetsto size 2.
� A covering of H is a collection of hyperedges such that everyelement of S is contained in some hyperedge.
� The covering number k(H) of a hypergraph H is the number ofhyperedges in the smallest covering of H.
� A t-fold covering of H is a collection of hyperedges such thatevery element of S is contained in at least t hyperedges.
� The t-fold covering number kt(H) is the number of hyperedgesin the smallest t-fold covering of H.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Definitions (Review)
� A hypergraph H is a pair (S ,X ) where S is a finite set and Xconsists of subsets of S , called hyperedges.
� Note that a graph is a hypergraph where we restrict the subsetsto size 2.
� A covering of H is a collection of hyperedges such that everyelement of S is contained in some hyperedge.
� The covering number k(H) of a hypergraph H is the number ofhyperedges in the smallest covering of H.
� A t-fold covering of H is a collection of hyperedges such thatevery element of S is contained in at least t hyperedges.
� The t-fold covering number kt(H) is the number of hyperedgesin the smallest t-fold covering of H.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
H(G )
Let H(G ) be the hypergraph whose vertices are the edges of G andwhose hyperedges are the maximal matchings of G .
Example:
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
H(G )
Let H(G ) be the hypergraph whose vertices are the edges of G andwhose hyperedges are the maximal matchings of G .
Example:
Matchings of G
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
H(G )
Let H(G ) be the hypergraph whose vertices are the edges of G andwhose hyperedges are the maximal matchings of G .
Example:
G and H(G )
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Fractional Covering Number (Review)
� Recall that the (t-fold) covering number is the number ofhyperedges in the smallest (t-fold) covering of H.
� We similarly define the fractional covering number kf (H) aslimt→∞
kt
t .
� Equivalently, kf = inftkt
t .
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Matchings
� Recall that a matching is a collection of edges such that novertex is incident to more than one edge in the matching. Thematching number, α′(G ), is the greatest number of edges in amatching of G .
� In integer programming terms, we let xe ∈ {0, 1} be theindicator variable that is 1 if e ∈ M and 0 otherwise. Then
α′(G ) = max∑
e
xe such that ∀v ,∑e3v
xe ≤ 1.
� To find the fractional matching number, α′∗(G ), simply relaxthe requirements to 0 ≤ xe and xe ≤ 1. This gives us the desiredlinear program.
� Note that since the fractional matching number has fewerrequirements and this is a maximum, α′(G ) ≤ α′∗(G ).
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Matchings
� Recall that a matching is a collection of edges such that novertex is incident to more than one edge in the matching. Thematching number, α′(G ), is the greatest number of edges in amatching of G .
� In integer programming terms, we let xe ∈ {0, 1} be theindicator variable that is 1 if e ∈ M and 0 otherwise. Then
α′(G ) = max∑
e
xe such that ∀v ,∑e3v
xe ≤ 1.
� To find the fractional matching number, α′∗(G ), simply relaxthe requirements to 0 ≤ xe and xe ≤ 1. This gives us the desiredlinear program.
� Note that since the fractional matching number has fewerrequirements and this is a maximum, α′(G ) ≤ α′∗(G ).
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Matchings
� Recall that a matching is a collection of edges such that novertex is incident to more than one edge in the matching. Thematching number, α′(G ), is the greatest number of edges in amatching of G .
� In integer programming terms, we let xe ∈ {0, 1} be theindicator variable that is 1 if e ∈ M and 0 otherwise. Then
α′(G ) = max∑
e
xe such that ∀v ,∑e3v
xe ≤ 1.
� To find the fractional matching number, α′∗(G ), simply relaxthe requirements to 0 ≤ xe and xe ≤ 1. This gives us the desiredlinear program.
� Note that since the fractional matching number has fewerrequirements and this is a maximum, α′(G ) ≤ α′∗(G ).
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Matchings
� Recall that a matching is a collection of edges such that novertex is incident to more than one edge in the matching. Thematching number, α′(G ), is the greatest number of edges in amatching of G .
� In integer programming terms, we let xe ∈ {0, 1} be theindicator variable that is 1 if e ∈ M and 0 otherwise. Then
α′(G ) = max∑
e
xe such that ∀v ,∑e3v
xe ≤ 1.
� To find the fractional matching number, α′∗(G ), simply relaxthe requirements to 0 ≤ xe and xe ≤ 1. This gives us the desiredlinear program.
� Note that since the fractional matching number has fewerrequirements and this is a maximum, α′(G ) ≤ α′∗(G ).
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Vertex Covers
� The fractional vertex cover number, β∗(G ), is simply the dual ofα′∗(G ). In other words, if 0 ≤ xv , xv ≤ 1,
β∗(G ) = min∑
v
xv such that ∀e,∑v∈e
xv ≥ 1.
� The traditional vertex cover number, β(G ) is the same thing,with the restriction that xv ∈ {0, 1}. Since this is a minimumand β has more restrictions, β∗(G ) ≤ β(G ).
� Note that LP duality gives us that α′∗ = β∗, so
α′ ≤ α′∗ = β∗ ≤ β.
� For bipartite graphs, the Konig-Egervary Theorem states thatα′ = β, so equality holds throughout.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Vertex Covers
� The fractional vertex cover number, β∗(G ), is simply the dual ofα′∗(G ). In other words, if 0 ≤ xv , xv ≤ 1,
β∗(G ) = min∑
v
xv such that ∀e,∑v∈e
xv ≥ 1.
� The traditional vertex cover number, β(G ) is the same thing,with the restriction that xv ∈ {0, 1}. Since this is a minimumand β has more restrictions, β∗(G ) ≤ β(G ).
� Note that LP duality gives us that α′∗ = β∗, so
α′ ≤ α′∗ = β∗ ≤ β.
� For bipartite graphs, the Konig-Egervary Theorem states thatα′ = β, so equality holds throughout.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Vertex Covers
� The fractional vertex cover number, β∗(G ), is simply the dual ofα′∗(G ). In other words, if 0 ≤ xv , xv ≤ 1,
β∗(G ) = min∑
v
xv such that ∀e,∑v∈e
xv ≥ 1.
� The traditional vertex cover number, β(G ) is the same thing,with the restriction that xv ∈ {0, 1}. Since this is a minimumand β has more restrictions, β∗(G ) ≤ β(G ).
� Note that LP duality gives us that α′∗ = β∗, so
α′ ≤ α′∗ = β∗ ≤ β.
� For bipartite graphs, the Konig-Egervary Theorem states thatα′ = β, so equality holds throughout.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Vertex Covers
� The fractional vertex cover number, β∗(G ), is simply the dual ofα′∗(G ). In other words, if 0 ≤ xv , xv ≤ 1,
β∗(G ) = min∑
v
xv such that ∀e,∑v∈e
xv ≥ 1.
� The traditional vertex cover number, β(G ) is the same thing,with the restriction that xv ∈ {0, 1}. Since this is a minimumand β has more restrictions, β∗(G ) ≤ β(G ).
� Note that LP duality gives us that α′∗ = β∗, so
α′ ≤ α′∗ = β∗ ≤ β.
� For bipartite graphs, the Konig-Egervary Theorem states thatα′ = β, so equality holds throughout.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Examples
� α′(C5) = 2
� α′∗(C5) = 2.5
� α′(C6) = α′∗(C6) = 3, since C6 is bipartite
� Note that in C5, the 1/2 matching isn’t a convex combination ofinteger matchings, but in C6, it is.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Examples
� α′(C5) = 2
� α′∗(C5) = 2.5
� α′(C6) = α′∗(C6) = 3, since C6 is bipartite
� Note that in C5, the 1/2 matching isn’t a convex combination ofinteger matchings, but in C6, it is.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Examples
� α′(C5) = 2
� α′∗(C5) = 2.5
� α′(C6) = α′∗(C6) = 3, since C6 is bipartite
� Note that in C5, the 1/2 matching isn’t a convex combination ofinteger matchings, but in C6, it is.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Examples
� α′(C5) = 2
� α′∗(C5) = 2.5
� α′(C6) = α′∗(C6) = 3, since C6 is bipartite
� Note that in C5, the 1/2 matching isn’t a convex combination ofinteger matchings, but in C6, it is.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Partitions into Matchings
� There are many ways to partition the edge set of a graph G intomatchings.
� Here are three such ways for the graph G from earlier.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Partitions into Matchings
� There are many ways to partition the edge set of a graph G intomatchings.
� Here are three such ways for the graph G from earlier.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Edge Chromatic Number
� The size of the smallest partition into matchings is called theedge chromatic number, χ′(G ).
� Recall that we usually think of χ′ as being the minimum numberof colors required to properly edge color a graph, but this is thesame thing, since if each part is assigned a color, a propercoloring results.
� Alternatively, the edge chromatic number is the covering numberof H(G ).
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Edge Chromatic Number
� The size of the smallest partition into matchings is called theedge chromatic number, χ′(G ).
� Recall that we usually think of χ′ as being the minimum numberof colors required to properly edge color a graph, but this is thesame thing, since if each part is assigned a color, a propercoloring results.
� Alternatively, the edge chromatic number is the covering numberof H(G ).
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Edge Chromatic Number
� The size of the smallest partition into matchings is called theedge chromatic number, χ′(G ).
� Recall that we usually think of χ′ as being the minimum numberof colors required to properly edge color a graph, but this is thesame thing, since if each part is assigned a color, a propercoloring results.
� Alternatively, the edge chromatic number is the covering numberof H(G ).
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Here, we need all three hyperedges to contain all vertices of H(G ), soχ′(G ) = 3.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Fractional Edge Chromatic Number
� The fractional edge chromatic number, χ′f (G ), is equal to thefractional covering number kf (H(G )).
� Equivalently, χ′f (G ) = χf (L(G )), where χf is the fractionalchromatic number and L(G ) is the line graph of G .
� Additionally,
χ′f (G ) = limt→∞
χ′t(G )
t.
� Finally, and most importantly, χ′f (G ) relates to valid fractionaledge colorings.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Fractional Edge Chromatic Number
� The fractional edge chromatic number, χ′f (G ), is equal to thefractional covering number kf (H(G )).
� Equivalently, χ′f (G ) = χf (L(G )), where χf is the fractionalchromatic number and L(G ) is the line graph of G .
� Additionally,
χ′f (G ) = limt→∞
χ′t(G )
t.
� Finally, and most importantly, χ′f (G ) relates to valid fractionaledge colorings.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Fractional Edge Chromatic Number
� The fractional edge chromatic number, χ′f (G ), is equal to thefractional covering number kf (H(G )).
� Equivalently, χ′f (G ) = χf (L(G )), where χf is the fractionalchromatic number and L(G ) is the line graph of G .
� Additionally,
χ′f (G ) = limt→∞
χ′t(G )
t.
� Finally, and most importantly, χ′f (G ) relates to valid fractionaledge colorings.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Fractional Edge Chromatic Number
� The fractional edge chromatic number, χ′f (G ), is equal to thefractional covering number kf (H(G )).
� Equivalently, χ′f (G ) = χf (L(G )), where χf is the fractionalchromatic number and L(G ) is the line graph of G .
� Additionally,
χ′f (G ) = limt→∞
χ′t(G )
t.
� Finally, and most importantly, χ′f (G ) relates to valid fractionaledge colorings.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Fractional Edge Colorings
� A fractional edge coloring assigns non-negative weights wM toeach matching M of G such that for each edge e ∈ E (G ),∑
M3e
wM ≥ 1.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Fractional Edge Colorings
� A fractional edge coloring assigns non-negative weights wM toeach matching M of G such that for each edge e ∈ E (G ),∑
M3e
wM ≥ 1.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Fractional Edge Colorings
� This leads to a picture like:
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Fractional Edge Chromatic Number
� Let w be a fractional edge coloring. Then the fractional edgechromatic number χ′f (G ) is equal to
minw
∑M
wM .
� By duality,
χ′f (G ) = maxw
∑e
we ,
where for each matching,∑
M3e
we ≤ 1.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Fractional Edge Chromatic Number
� Let w be a fractional edge coloring. Then the fractional edgechromatic number χ′f (G ) is equal to
minw
∑M
wM .
� By duality,
χ′f (G ) = maxw
∑e
we ,
where for each matching,∑
M3e
we ≤ 1.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Theorem 2.1.5
For any graph G , 2α′∗ is an integer. Moreover, there exists afractional matching f for which∑
e∈E(G)
f (e) = α′∗(G )
such that f (e) ∈ {0, 1/2, 1} for every edge e.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Proof of Theorem 2.1.5
� Of all matchings f such that∑
e f (e) = α′∗, choose one withthe greatest number of edges such that f (e) = 0. Let H be thesubgraph induced on the set of edges with nonzero values.
� We next prove several claims about H.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Proof of Theorem 2.1.5
� Of all matchings f such that∑
e f (e) = α′∗, choose one withthe greatest number of edges such that f (e) = 0. Let H be thesubgraph induced on the set of edges with nonzero values.
� We next prove several claims about H.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Claim 1
� Claim: H has no even cycles.
� Proof:� Suppose for the sake of contradiction that there exists an even
cycle C . There must be some edge with minimal value; supposethat edge is e0 and f (e0) = m.
� Define g : E(G)→ {−1, 0, 1} such that g(e0) = −1, galternates between −1 and 1 around C , and g is 0 on everyother edge of G .
� Then f + mg is a new fractional matching with the same sumand one more 0 edge, which is a contradiction.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Claim 1
� Claim: H has no even cycles.
� Proof:� Suppose for the sake of contradiction that there exists an even
cycle C . There must be some edge with minimal value; supposethat edge is e0 and f (e0) = m.
� Define g : E(G)→ {−1, 0, 1} such that g(e0) = −1, galternates between −1 and 1 around C , and g is 0 on everyother edge of G .
� Then f + mg is a new fractional matching with the same sumand one more 0 edge, which is a contradiction.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Claim 1
� Claim: H has no even cycles.
� Proof:� Suppose for the sake of contradiction that there exists an even
cycle C . There must be some edge with minimal value; supposethat edge is e0 and f (e0) = m.
� Define g : E(G)→ {−1, 0, 1} such that g(e0) = −1, galternates between −1 and 1 around C , and g is 0 on everyother edge of G .
� Then f + mg is a new fractional matching with the same sumand one more 0 edge, which is a contradiction.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Claim 1
� Claim: H has no even cycles.
� Proof:� Suppose for the sake of contradiction that there exists an even
cycle C . There must be some edge with minimal value; supposethat edge is e0 and f (e0) = m.
� Define g : E(G)→ {−1, 0, 1} such that g(e0) = −1, galternates between −1 and 1 around C , and g is 0 on everyother edge of G .
� Then f + mg is a new fractional matching with the same sumand one more 0 edge, which is a contradiction.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Claim 2
� Claim: H has no pendant edges, unless they’re entirecomponents of H.
� Proof: Suppose e = uv were a pendant edge where d(u) > 1and d(v) = 1.
� If f (e) = 1, then e is an entire component of H, since no edgetouching it can have value other than 0.
� If f (e) < 1, then increasing f (e) to 1 and decreasing the valueof f on all edges incident to u to 0 creates a matching with thesame sum but more 0 edges, a contradiction.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Claim 2
� Claim: H has no pendant edges, unless they’re entirecomponents of H.
� Proof: Suppose e = uv were a pendant edge where d(u) > 1and d(v) = 1.
� If f (e) = 1, then e is an entire component of H, since no edgetouching it can have value other than 0.
� If f (e) < 1, then increasing f (e) to 1 and decreasing the valueof f on all edges incident to u to 0 creates a matching with thesame sum but more 0 edges, a contradiction.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Claim 2
� Claim: H has no pendant edges, unless they’re entirecomponents of H.
� Proof: Suppose e = uv were a pendant edge where d(u) > 1and d(v) = 1.
� If f (e) = 1, then e is an entire component of H, since no edgetouching it can have value other than 0.
� If f (e) < 1, then increasing f (e) to 1 and decreasing the valueof f on all edges incident to u to 0 creates a matching with thesame sum but more 0 edges, a contradiction.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Claim 2
� Claim: H has no pendant edges, unless they’re entirecomponents of H.
� Proof: Suppose e = uv were a pendant edge where d(u) > 1and d(v) = 1.
� If f (e) = 1, then e is an entire component of H, since no edgetouching it can have value other than 0.
� If f (e) < 1, then increasing f (e) to 1 and decreasing the valueof f on all edges incident to u to 0 creates a matching with thesame sum but more 0 edges, a contradiction.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Claim 3
� Claim: If H has an odd cycle C , it must be an entirecomponent of H.
� Proof: Suppose C is an odd cycle and suppose v ∈ C is suchthat dH(v) ≥ 3.
� Start at v , choose an edge not in C , and trace a longest path inH. Note that this path can’t return to C , since that wouldcreate an even cycle.
� The path also can’t reach a vertex of degree 1, since there aren’tany pendant edges.
� Therefore, it ends at a vertex such that any other edge revisitsthe path, which would form another cycle.
� Thus, H contains a subset K that consists of two odd cyclesconnected by a path, where the path may have length 0.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Claim 3
� Claim: If H has an odd cycle C , it must be an entirecomponent of H.
� Proof: Suppose C is an odd cycle and suppose v ∈ C is suchthat dH(v) ≥ 3.
� Start at v , choose an edge not in C , and trace a longest path inH. Note that this path can’t return to C , since that wouldcreate an even cycle.
� The path also can’t reach a vertex of degree 1, since there aren’tany pendant edges.
� Therefore, it ends at a vertex such that any other edge revisitsthe path, which would form another cycle.
� Thus, H contains a subset K that consists of two odd cyclesconnected by a path, where the path may have length 0.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Claim 3
� Claim: If H has an odd cycle C , it must be an entirecomponent of H.
� Proof: Suppose C is an odd cycle and suppose v ∈ C is suchthat dH(v) ≥ 3.
� Start at v , choose an edge not in C , and trace a longest path inH. Note that this path can’t return to C , since that wouldcreate an even cycle.
� The path also can’t reach a vertex of degree 1, since there aren’tany pendant edges.
� Therefore, it ends at a vertex such that any other edge revisitsthe path, which would form another cycle.
� Thus, H contains a subset K that consists of two odd cyclesconnected by a path, where the path may have length 0.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Claim 3
� Claim: If H has an odd cycle C , it must be an entirecomponent of H.
� Proof: Suppose C is an odd cycle and suppose v ∈ C is suchthat dH(v) ≥ 3.
� Start at v , choose an edge not in C , and trace a longest path inH. Note that this path can’t return to C , since that wouldcreate an even cycle.
� The path also can’t reach a vertex of degree 1, since there aren’tany pendant edges.
� Therefore, it ends at a vertex such that any other edge revisitsthe path, which would form another cycle.
� Thus, H contains a subset K that consists of two odd cyclesconnected by a path, where the path may have length 0.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Claim 3
� Claim: If H has an odd cycle C , it must be an entirecomponent of H.
� Proof: Suppose C is an odd cycle and suppose v ∈ C is suchthat dH(v) ≥ 3.
� Start at v , choose an edge not in C , and trace a longest path inH. Note that this path can’t return to C , since that wouldcreate an even cycle.
� The path also can’t reach a vertex of degree 1, since there aren’tany pendant edges.
� Therefore, it ends at a vertex such that any other edge revisitsthe path, which would form another cycle.
� Thus, H contains a subset K that consists of two odd cyclesconnected by a path, where the path may have length 0.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Claim 3
� Claim: If H has an odd cycle C , it must be an entirecomponent of H.
� Proof: Suppose C is an odd cycle and suppose v ∈ C is suchthat dH(v) ≥ 3.
� Start at v , choose an edge not in C , and trace a longest path inH. Note that this path can’t return to C , since that wouldcreate an even cycle.
� The path also can’t reach a vertex of degree 1, since there aren’tany pendant edges.
� Therefore, it ends at a vertex such that any other edge revisitsthe path, which would form another cycle.
� Thus, H contains a subset K that consists of two odd cyclesconnected by a path, where the path may have length 0.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Claim 3 (continued)
� H contains a subset K that consists of two odd cycles connectedby a path, where the path may have length 0.
� Let g assign 0 to edges not on K , alternate 1/2 and −1/2around the cycles, and alternate 1 and −1 on the path betweenthem. Note that this sums to 0.
� Let m ∈ R have the smallest absolute value such thatf + mg = 0 on an edge of K . Then f + mg is a matching withmore 0 edges than f , a contradiction.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Claim 3 (continued)
� H contains a subset K that consists of two odd cycles connectedby a path, where the path may have length 0.
� Let g assign 0 to edges not on K , alternate 1/2 and −1/2around the cycles, and alternate 1 and −1 on the path betweenthem. Note that this sums to 0.
� Let m ∈ R have the smallest absolute value such thatf + mg = 0 on an edge of K . Then f + mg is a matching withmore 0 edges than f , a contradiction.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Claim 3 (continued)
� H contains a subset K that consists of two odd cycles connectedby a path, where the path may have length 0.
� Let g assign 0 to edges not on K , alternate 1/2 and −1/2around the cycles, and alternate 1 and −1 on the path betweenthem. Note that this sums to 0.
� Let m ∈ R have the smallest absolute value such thatf + mg = 0 on an edge of K . Then f + mg is a matching withmore 0 edges than f , a contradiction.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Summary of Claims
� Therefore, H consists only of components that are odd cycles orcopies of K2.
� On copies of K2, f = 1, and on odd cycles, f = 1/2. Sincef = 0 on the rest of G , f only takes on values in {0, 1/2, 1}.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Summary of Claims
� Therefore, H consists only of components that are odd cycles orcopies of K2.
� On copies of K2, f = 1, and on odd cycles, f = 1/2. Sincef = 0 on the rest of G , f only takes on values in {0, 1/2, 1}.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Fractional Transversals
� A fractional transversal of G is a function g : V (G )→ [0, 1]such that
∑v∈e g(v) ≥ 1 for every e ∈ E (G ).
� The fractional transversal number is the infimum of∑v∈V (G) g(v) over all fractional transversals g . By duality, this
is α′∗(G ).
� Theorem 2.1.6: For every graph G , there exists a fractionaltransversal g for which∑
v∈V (G)
g(v) = α′∗(G )
such that g(v) ∈ {0, 1/2, 1} for every vertex v .
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Fractional Transversals
� A fractional transversal of G is a function g : V (G )→ [0, 1]such that
∑v∈e g(v) ≥ 1 for every e ∈ E (G ).
� The fractional transversal number is the infimum of∑v∈V (G) g(v) over all fractional transversals g . By duality, this
is α′∗(G ).
� Theorem 2.1.6: For every graph G , there exists a fractionaltransversal g for which∑
v∈V (G)
g(v) = α′∗(G )
such that g(v) ∈ {0, 1/2, 1} for every vertex v .
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Fractional Transversals
� A fractional transversal of G is a function g : V (G )→ [0, 1]such that
∑v∈e g(v) ≥ 1 for every e ∈ E (G ).
� The fractional transversal number is the infimum of∑v∈V (G) g(v) over all fractional transversals g . By duality, this
is α′∗(G ).
� Theorem 2.1.6: For every graph G , there exists a fractionaltransversal g for which∑
v∈V (G)
g(v) = α′∗(G )
such that g(v) ∈ {0, 1/2, 1} for every vertex v .
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Perfect Matchings
� A perfect matching is a set of edges such that every vertex is insome edge of the matching.
� A fractional perfect matching is a fractional matching f suchthat
∑e f (e) = |V (G )|/2.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Perfect Matchings
� A perfect matching is a set of edges such that every vertex is insome edge of the matching.
� A fractional perfect matching is a fractional matching f suchthat
∑e f (e) = |V (G )|/2.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Tutte’s Theorem
� Let o(G ) be the number of odd components and i(G ) thenumber of isolated vertices.
� Tutte’s Theorem: A graph G has a perfect matching if andonly if o(G − S) ≤ |S | for all S ⊆ V (G ).
� Fractional Tutte’s Theorem: A graph G has a fractionalperfect matching if and only if i(G − S) ≤ |S | for all S ⊆ V (G ).
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Tutte’s Theorem
� Let o(G ) be the number of odd components and i(G ) thenumber of isolated vertices.
� Tutte’s Theorem: A graph G has a perfect matching if andonly if o(G − S) ≤ |S | for all S ⊆ V (G ).
� Fractional Tutte’s Theorem: A graph G has a fractionalperfect matching if and only if i(G − S) ≤ |S | for all S ⊆ V (G ).
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Tutte’s Theorem
� Let o(G ) be the number of odd components and i(G ) thenumber of isolated vertices.
� Tutte’s Theorem: A graph G has a perfect matching if andonly if o(G − S) ≤ |S | for all S ⊆ V (G ).
� Fractional Tutte’s Theorem: A graph G has a fractionalperfect matching if and only if i(G − S) ≤ |S | for all S ⊆ V (G ).
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Bounds
Consider some quick bounds on χ′(G ).
� χ′(G ) ≥ ∆(G )
� χ′(G ) ≥
⌈e(G )
bv(G )/2c
⌉
Proof.
Each color class is a matching, and matchings have at mostbv(G )/2c edges. Apply the pigeonhole principle.
� We can improve this slightly to get χ′(G ) ≥ maxH
⌈e(H)
bv(H)/2c
⌉,
where H is a subgraph of G . This is because χ′(G ) ≥ χ′(H).
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Bounds
Consider some quick bounds on χ′(G ).
� χ′(G ) ≥ ∆(G )
� χ′(G ) ≥
⌈e(G )
bv(G )/2c
⌉
Proof.
Each color class is a matching, and matchings have at mostbv(G )/2c edges. Apply the pigeonhole principle.
� We can improve this slightly to get χ′(G ) ≥ maxH
⌈e(H)
bv(H)/2c
⌉,
where H is a subgraph of G . This is because χ′(G ) ≥ χ′(H).
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Bounds
Consider some quick bounds on χ′(G ).
� χ′(G ) ≥ ∆(G )
� χ′(G ) ≥
⌈e(G )
bv(G )/2c
⌉
Proof.
Each color class is a matching, and matchings have at mostbv(G )/2c edges. Apply the pigeonhole principle.
� We can improve this slightly to get χ′(G ) ≥ maxH
⌈e(H)
bv(H)/2c
⌉,
where H is a subgraph of G . This is because χ′(G ) ≥ χ′(H).
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Bounds
Consider some quick bounds on χ′(G ).
� χ′(G ) ≥ ∆(G )
� χ′(G ) ≥
⌈e(G )
bv(G )/2c
⌉
Proof.
Each color class is a matching, and matchings have at mostbv(G )/2c edges. Apply the pigeonhole principle.
� We can improve this slightly to get χ′(G ) ≥ maxH
⌈e(H)
bv(H)/2c
⌉,
where H is a subgraph of G . This is because χ′(G ) ≥ χ′(H).
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Bounds
In searching for a subgraph H that achieves the maximum, we may
assume that |V (H)| is odd. To see why, suppose maxH
⌈e(H)
bv(H)/2c
⌉is achieved by H where |V (H)| is even and let v ∈ V (H) haveminimum degree.
Then
δ(H) ≤ 2|E (H)||V (H)|
,
|E (H)| − δ(H)12 (|V (H)| − 2)
≥ 2|E (H)||V (H)|
,
|E (H − v)|b|V (H − v)|/2c
≥ |E (H)|b|V (H)|/2c
.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Bounds
In searching for a subgraph H that achieves the maximum, we may
assume that |V (H)| is odd. To see why, suppose maxH
⌈e(H)
bv(H)/2c
⌉is achieved by H where |V (H)| is even and let v ∈ V (H) haveminimum degree. Then
δ(H) ≤ 2|E (H)||V (H)|
,
|E (H)| − δ(H)12 (|V (H)| − 2)
≥ 2|E (H)||V (H)|
,
|E (H − v)|b|V (H − v)|/2c
≥ |E (H)|b|V (H)|/2c
.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Bounds
� Therefore, we may assume H is induced and has an odd numberof vertices. We may also suppose there are at least threevertices.
� Define
Λ(G ) = maxH
2|E (H)||V (H)| − 1
,
where H is as stated previously. (For |V (G )| < 3, let Λ(G ) = 0.)
� Then χ′(G ) ≥ max{∆(G ), dΛ(G )e}. The same holds for χ′t/t,and thus χ′f (G ) ≥ max{∆(G ),Λ(G )}.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Bounds
� Therefore, we may assume H is induced and has an odd numberof vertices. We may also suppose there are at least threevertices.
� Define
Λ(G ) = maxH
2|E (H)||V (H)| − 1
,
where H is as stated previously. (For |V (G )| < 3, let Λ(G ) = 0.)
� Then χ′(G ) ≥ max{∆(G ), dΛ(G )e}. The same holds for χ′t/t,and thus χ′f (G ) ≥ max{∆(G ),Λ(G )}.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Bounds
� Therefore, we may assume H is induced and has an odd numberof vertices. We may also suppose there are at least threevertices.
� Define
Λ(G ) = maxH
2|E (H)||V (H)| − 1
,
where H is as stated previously. (For |V (G )| < 3, let Λ(G ) = 0.)
� Then χ′(G ) ≥ max{∆(G ), dΛ(G )e}. The same holds for χ′t/t,and thus χ′f (G ) ≥ max{∆(G ),Λ(G )}.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Fractional Chromatic Number
In fact,χ′f (G ) = max{∆(G ),Λ(G )}.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Vizing’s Theorem
� Vizing’s Theorem: For any graph G ,∆(G ) ≤ χ′(G ) ≤ ∆(G ) + 1.
� Fractional Vizing’s Theorem: For any graph G ,∆(G ) ≤ χ′f (G ) ≤ ∆(G ) + 1.
Fractional EdgeColoring
Debbie Berg
Fractional EdgeChromaticNumber
Hypergraphs andCoveringNumbers
Matchings
Edge ChromaticNumber
Fractional EdgeColorings
Theorems
Transversals andMatchings
Bounds on χ′and χ′
f
Vizing’s Theorem
� Vizing’s Theorem: For any graph G ,∆(G ) ≤ χ′(G ) ≤ ∆(G ) + 1.
� Fractional Vizing’s Theorem: For any graph G ,∆(G ) ≤ χ′f (G ) ≤ ∆(G ) + 1.