Longest Cycles in 3-Connected Graphs Contain Three Contractible Edges

- Nathaniel Dean

BELL COMMUNICATIONS RESEARCH MORRISTOWN, NEW JERSEY USA

Robert L. Hemminger

Katsuhiro Ota

VANDERBILT UNlVERSITY NASHVILLE, TENNESSEE USA

UNIVERSITY OF TOKYO TOKYO, JAPAN

ABSTRACT

We show that if G is a 3-connected graph of order at least seven, then every longest path between distinct vertices in G contains at least two contractible edges. An immediate corollary is that longest cycles in such graphs contain at least three contractible edges.

We consider only finite undirected simple graphs. An edge of a 3-connected graph is called contractible if its contraction results in a 3-connected graph; otherwise it is called noncontractible. Note that in a 3-connected graph of order at least five, an edge uu is noncontractible if and only if there exists a 3-cutset containing both u and u. We call such a 3-cutset one associated with the edge uu. Other terminology is as in [2].

It easily follows from results of Tutte [5 ] that every 3-connected graph on at least five vertices contains a contractible edge. Thomassen [4] gave a simpler proof of this (and then used it in his elegant induction proof of Kuratowski’s Theorem). Using methods similar to Thomassen’s, Dean, Hemminger, and Toft [3] improved on this by showing that, in the same setting, longest (x,y)-paths contain at least one contractible edge if xy is an edge. In this paper, we use dif- ferent methods to get the best result of this type; namely, with two small excep-

Journal of Graph Theory, Vol. 13, No. 1, 17-21 (1989) 0 1989 by John Wiley & Sons, Inc. CCC 0364-9024/89/010017-05$04.00

18 JOURNAL OF GRAPH THEORY

tions, every longest (x,y)-path contains at least two contractible edges. As an immediate corollary, we have that every longest cycle contains at least three contractible edges.

These results were found independently by the first two authors and the third. We first prove two lemmas. In these, let G be a 3-connected graph of order

at least five (that is, not K4) , let x and y be distinct vertices of G, and let P : x = x , , x 2 , x3, . . . , x, = y be a longest ( x , y)-path in G.

Lemma 1. If xixi+l is a noncontractible edge and S = { x i , x i + , , w} is an asso- ciated 3-cutset, then every component of G - S contains a vertex of P. More- over, w € V(P) - {x,y} if i = l or n - l .

Proof. Let A be a component of G - S that P does not meet. Since xi and xi+l are both adjacent to vertices in A , we get a longer (x, y)-path than P by re- placing the edge x ix i t l by an (x,,x,,,)-path through A.

If i = 1 or n - I and w $E V(P) - {x,y}, then P - S is connected, which contradicts the first part of the lemma. I

Lemma 2. If x ix i+ l is a noncontractible edge, S = {+xi+,, w} is an associ- ated 3-cutset, and A is a component of G - S that contains neither x nor y, then P contains a contractible edge with at least one endvertex in A.

Proof. We can assume that i, S, and A were chosen so that A contains no subgraph obtained in the same manner as A . By Lemma 1, we can only get such an A if w E V(P). Let wu be the unique (w,A)-edge of P and assume that wu is noncontractible with associated 3-cutset T = {u, w, t } .

Let B be any component of G - S other than A . Then G' = G[V(B) U {x,,xi+,}] is 2-connected (if z were a cutvertex of G' , then {w,z} would be a 2-cutset of G). Then K = G - V(A) - {w} is 2-connected since it is the union of 2-connected subgraphs containing a common edge. So K - T = K - { t } is a connected subgraph of G - T. Hence, some component of G - T is a proper subgraph of A , contradicting the assumed minimality of A . I

Corollary 3. P contains a contractible edge.

Proof. Suppose x I x z is noncontractible. Then, from Lemma 1 , we get w E V(P). Hence P - S only has two components, and so G - S has a corn- ponent containing neither x nor y. It follows from Lemma 2 that P contains a contractible edge. I

Theorem 4. If G is a 3-connected graph other than K4, K2 X K 3 , or the com- plement of a path on six vertices (see Figure l ) , x and y are distinct vertices of G , and P is a longest (x,y)-path, then P contains at least two contractible edges.

LONGEST CYCLES IN 3-CONNECTED GRAPHS 19

FIGURE 1

Proof. We assume that G has at least five vertices (since G 9 K4) and that P contains only one contractible edge, say X ~ X ~ + ~ . From this we will show that G must be one of the other two exceptional graphs.

Supposing that k = n - 1 makes x I x 2 noncontractible and so, as in the proof of Corollary 3, P has another contractible edge. Thus, by symmetry, we have 2 5 k 5 n - 2.

So let S = { x I , x 2 , s} and T = { X , - ~ , X , , , t } be 3-cutsets associated with the edges x I x 2 and X,-~X,, respectively. By Lemmas 1 and 2, s = x j E {xk+I, . . . , with s # X 3 and t = Xi E { X 2 , . . . , xk} with t # X,-2. Let A be the component of G - S that contains xj and let B be the component of G - T that contains x,+. Hence, since P - S and P - T each have only two compo- nents, P n A = {x3 , . . . ,x i - , } and P fl B = {xi+, , . . . , x , - ~ } (see Figure 2) .

Let A* be the subgraph of G induced by V ( G ) - S - V(A) and let B * be the subgraph of G induced by V ( G ) - T - V(B).

Now there are no edges between A* and A , so there are no edges from x, to A. But there must be an edge from x, to B since {x , , -~ , t } is not a cutset of G . Hence j < n - 1, that is, s # x , - ~ . By symmetry, t # x2. Thus t E V(A) and s E V ( B ) . Consequently, {V(B*) , V(A n B ) , V(A*) , { s , r } } partitions V ( G ) . Hence A fl B is a component of G - {s, t } if V(A f l B ) # 0, so we must have V(A fl B ) = 0. It follows that t = xk , s = and V(A) - {t} G V(B*). But this, s E V(B) , and V(A) - { t } # 0 imply that A - { t ) is a component of G - { x l , x 2 , t } , in contradiction of Lemma 2, Thus V(A) - ( t } = 0 and, by symmetry, V(B) - {s} = 0.

Therefore G is a graph on six vertices, dg(s) = dg(t) = 3 and st is the only contractible edge in P. Thus G - { x 2 , t } has a cutvertex, so x , is adjacent to only one vertex of the set {x , , -~ , x,}. Similarly x, is adjacent to only one vertex of the set { x l , x 2 } . Hence G is one of the graphs in Figure 1 other than K4 and

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with x and y at the bottom and s and t at the top. (Note that K2 X K3 is obtained twice this way, once with x2x5 an edge and once when it is not an edge.)

Corollary 5. K2 X K3 contains at least three contractible edges.

Every longest cycle in a 3-connected graph other than K4 and

Proof. Let C be a longest cycle in such a graph G , and let xy E E(C) . One easily checks that P: is not an exception here. Thus, since C - xy is a

longest (x , y)-path, the theorem implies that it contains at least two contractible edges, say e and f. Likewise, C - e contains two contractible edges, and at least one of these must be different from e and f. I

The following two families of examples, due to Dean, Hemminger and TofL [3], show that these results are best possible concerning the number of con- tractible edges that can be contained in longest cycles or longest (x, y)-paths in 3-connected graphs. Let G, = (P,)’ + xy where P, is the path xI,xz,. . . ,x, with x = xl and y = x,, and let H , = G, - x2x4 - x3x5 + x2x5. Then G, and H , with n 2 7 have the desired properties; namely, they contain Hamilton paths (cycles) having only two (three) contractible edges. Also note that H6 = K2 X K3 and G6 = P i .

The authors are indebted to a referee for pointing out that we can say much more if G is minimally .?-connected (i.e., if G - e is not 3-connected for each edge e of G).

Theorem 6. If C is a cycle of length at least four in a minimally 3-connected graph G other than K4, then no three consecutive edges of C are noncontractible.

Proof. Suppose, on the contrary, that tu, uu, and uw are three consecutive edges of C that are noncontractible. If u and u are both of degree at least four, Tutte [5] showed that either uu is contractible or G - uu is 3-connected. It fol- lows that at least one of u or u, say u, is of degree three. Thus, by a result of Ando, Enomoto, and Saito [l], uw E E(G) and dg(u) = dg(w) = 3. Hence uw is also noncontractible (since u is of degree two after uw is contracted). But in [ l ] it is also shown that there is at least one contractible edge incident with a vertex of degree three-which is contradicted at u. I

Corollary 7. graph other than K4, then at least a third of the edges of C are contractible.

manner) shows that this bound is best possible.

If C is a cycle of length at least four in a minimally 3-connected

Replacing each vertex on the rim of a wheel by a triangle (in the usual cubic

References

[l] K. Ando, H. Enomoto, and A. Saito, Contractible edges in 3-connected graphs. J . Combinat. Theory Ser. B 42 (1987) 87-93.

LONGEST CYCLES IN 3-CONNECTED GRAPHS 21

[2] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, Ameri- can Elsevier, New York, 1976.

[3] N. Dean, R . L . Hemminger, and B . Toft, On contractible edges in 3-connected graphs, Proceedings of the Eighteenth Southeastern Confer- ence on Combinatorics, Graph Theory and Computing, Boca Raton, FL (1987) Congressus Numerantium 58 (1987) 291-293.

[4] C. Thomassen, Planarity and duality of finite graphs, J . Combinat. Theory Ser. B 29 (1980) 224-271.

[5] W. T. Tutte, A theory of 3-connected graphs, Zndug. Math. 23 (1961) 441-445.