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Edge-colorings of cubic graphs with elements ofpoint-transitive Steiner triple systems
Daniel Kral’ 1,4
Institute for Theoretical Computer Science (ITI)Faculty of Mathematics and Physics, Charles UniversityMalostranske namestı 25, 118 00 Prague, Czech Republic
Edita Macajova 2,5
Department of Computer ScienceFaculty of Mathematics, Physics and Informatics, Comenius University
Mlynska dolina, 842 48 Bratislava, Slovakia
Attila Por 6
Department of Applied MathematicsFaculty of Mathematics and Physics, Charles UniversityMalostranske namestı 25, 118 00 Prague, Czech Republic
Jean-Sebastien Sereni 3,7
Department of Applied MathematicsFaculty of Mathematics and Physics, Charles UniversityMalostranske namestı 25, 118 00 Prague, Czech Republic
Abstract
A cubic graph G is S-edge-colorable for a Steiner triple system S if its edges canbe colored with points of S in such a way that the points assigned to three edges
Electronic Notes in Discrete Mathematics 29 (2007) 23–27
1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved.
www.elsevier.com/locate/endm
doi:10.1016/j.endm.2007.07.005
sharing a vertex form a triple in S. We show that a cubic graph is S-edge-colorablefor every non-trivial affine Steiner triple system S unless it contains a well-definedobstacle called a bipartite end. In addition, we show that all cubic graphs areS-edge-colorable for every non-projective non-affine point-transitive Steiner triplesystem S.
Keywords: Steiner triple systems, cubic graphs, edge-colorings.
1 Introduction
Edge-colorings of cubic (bridgeless) graphs form a prominent topic in graphtheory because of its close relation to deep and important problems such as theFour Color Theorem, the Cycle Double Cover Conjecture and many others.By Vizing’s theorem [7], the edges of any cubic graph can be colored withthree or four colors in such a way that the edges meeting at the same vertexreceive distinct colors. Non-trivially connected cubic graphs such that theiredges cannot be colored with three colors are called snarks.
Archdeacon [1] proposed to study edge-colorings of cubic graphs by pointsof Steiner triple systems since Steiner triple systems seem to be general enoughto “edge-color” most of cubic graphs and still well-structured enough to pro-vide us with new results on cubic graphs.
A Steiner triple system S is formed by n points and several triples suchthat every two distinct points are contained in exactly one common triple. Aclassical result asserts the existence of a Steiner triple system with n pointswhenever n = 1, 3 (mod 6), n ≥ 3. There are two prominent classes ofSteiner triple systems, projective and affine Steiner triple systems. A projectiveSteiner triple system PG(d, 2) is the Steiner triple system with 2d+1−1 pointscorresponding to non-zero (d + 1)-dimensional vectors over Z2 for d ≥ 1.Three such vectors form a triple of PG(d, 2) if they sum to the zero vector.The smallest Steiner triple system is the projective system PG(1, 2), the trivialSteiner triple system. The smallest non-trivial Steiner triple system, denoted
1 Institute for Theoretical computer science is supported as project 1M0545 by CzechMinistry of Education.2 This research was supported by VEGA grant no. 1/3022/06.3 This author has been supported by EU project Aeolus.4 Email: [email protected] Email: [email protected] Email: [email protected] Email: [email protected]
D. Král’ et al. / Electronic Notes in Discrete Mathematics 29 (2007) 23–2724
by S7, is PG(2, 2), the Fano plane. An affine Steiner triple system AG(d, 3)is the Steiner triple system with 3d points corresponding to d-dimensionalvectors over Z3 for d ≥ 1. Three such vectors form a triple of AG(d, 3) if theysum to the zero vector.
We now switch back to edge-colorings of cubic graphs. The points of aSteiner triple system S should be assigned to the edges of a cubic graph G insuch a way that the edges incident with the same vertex are assigned threedistinct points that form a triple of S. Edge-colorings with this propertyare called S-edge-colorings and G is said to be S-edge-colorable. A naturalquestion is for which cubic graphs G and which Steiner triple systems S, thereexists an S-edge-coloring of G.
Grannel et al. [3] established the existence of a Steiner triple system S with381 points such that every cubic graph (bridgeless or not) is S-edge-colorable(a system with this property is called universal). Later, Pal and Skoviera [6]showed that there exists a universal Steiner triple system with 21 points. Oneof the corollaries of our results is the existence of a universal Steiner triplesystem with 13 points (no Steiner triple system with less than 13 points canbe universal [4]).
Let us now survey further results in this area. Fu [2] showed that everycubic bridgeless graph of order less than 190 and those of genus at most24 are S7-edge-colorable. A stronger result was obtained by Holroyd andSkoviera [4] who showed that a cubic graph G is S-edge-colorable for a non-trivial projective Steiner triple system S if and only if it is bridgeless. Inparticular, all cubic bridgeless graphs are S7-edge-colorable. The condition onG being bridgeless can be easily seen to be necessary since an edge-coloring ofG with the points of a projective Steiner triple system PG(d, 2) can be viewedas a nowhere-zero Z
d+12 -flow.
A characterization of cubic graphs that are S-edge-colorable for a Steinertriple system S has been offered as a conjecture by Holroyd et al. [4]. One ofthe obstacles for the existence of an egde-coloring is the presence of a bipartiteend. If a cubic graph G has bridges it can be split along its bridges into 2-connected blocks, each incident with one or more bridges. Each bridge is splitinto two half-edges, each half-edge incident with one of the blocks. The blockincident with a single half-edge is called an end. Let H be an end of a cubicgraph and H ′ the graph obtained from H by suppressing the vertex incidentwith the half-edge. We say that H is a bipartite end if the graph H ′ is bipartite.
We can now state the conjectured characterization of cubic graphs thatare S-edge-colorable with a Steiner triple system S.
D. Král’ et al. / Electronic Notes in Discrete Mathematics 29 (2007) 23–27 25
Conjecture 1.1 (Holroyd and Skoviera [4], Conjecture 1.4) Let S be anon-projective Steiner triple system. A cubic graph G is S-edge-colorable un-less G has a bipartite end and S is affine.
If a cubic graph G has a bipartite end H and a Steiner triple system S is affine,an easy linear algebra argument [4] yields that the two edges of H incidentwith the bridge must be colored with the same point of S. Hence, G cannotbe S-edge-colored. The conjecture of Holroyd and Skoviera asserts this to bethe only obstacle for the existence of an S-edge-coloring unless S is projective.
2 Our contribution
We characterized [5] affine Steiner triple systems and Hall triple systems byforbidden configurations. Using these results, we prove Conjecture 1.1 withan additional assumption that the Steiner triple system S is point-transitive,i.e., for every two points x and y of S, there exists an automorphism of S thatmaps x to y. In fact, for non-trivial point-transitive Steiner triple systems S,we can characterize cubic graphs G that are S-edge-colorable.
Theorem 2.1 Let G be a cubic graph and S a non-trivial point-transitiveSteiner triple system.
• If S is projective, then G is S-edge-colorable if and only if G is bridgeless;
• if S is affine, G is S-edge-colorable if and only if G has no bipartite end;
• if S is neither projective nor affine, then G is always S-edge-colorable.
The first case in Theorem 2.1, the case of projective Steiner triple systems,follows from the results of Holroyd et al. [4]. The case of affine Steiner triplesystem provides a solution of a problem from [4,6] where the conjecture thatevery cubic graph with no bipartite end is AG(2, 3)-edge-colorable appeared.
Since there exists a point-transitive Steiner triple system with 13 pointsthat is neither projective nor affine, our results yield the existence of a uni-versal 13-point Steiner triple system. As no Steiner triple system with lessthan 13 points is universal, we provide an answer to a problem of Grannelet al. [3] to determine the number of points of the smallest universal Steinertriple system.
3 Concluding remarks
Conjecture 1.1 is now proven for all point-transitive Steiner triple systems, inparticular, for all affine non-trivial Steiner triple systems. Hence, it is only
D. Král’ et al. / Electronic Notes in Discrete Mathematics 29 (2007) 23–2726
open whether every non-projective non-affine Steiner triple system which isnot point-transitive is universal.
There are two Steiner triple system with 13 points: the point-transitivesystem S13 and the system S ′
13 that is not point-transitive. The system S13
is universal by our results. Along the lines of our proof, we tried to establishthat S ′
13 is also universal. First, we identified several configurations such thatif all of them are contained in S ′
13, all cubic graphs can be S ′13-edge-colored.
Next, we verified using a computer the presence of these configurations in S ′13
and concluded that the system S ′13 is universal. We believe that an analogous
argument can be used to establish that other non-point-transitive Steiner triplesystems are universal.
Acknowledgement
The second author would like to thank Department of Applied Mathematics ofCharles University for hospitality during her stay in Prague when this researchwas conducted. All the authors are indebted to Zdenek Dvorak for valuablediscussions on edge-colorings of cubic graphs.
References
[1] D. Archdeacon, Generalizations of Tait coloring cubic graphs,http://www.emba.uvm.edu/~archdeac/problems/problems.html.
[2] H.-L. Fu, A generalization of Tait coloring cubic graphs, Bull. Inst. Combin.Appl. 31 (2001), 45–49.
[3] M. J. Grannel, T. S. Griggs, M. Knor, and M. Skoviera, A Steiner triple systemwhich colours all cubic graphs, J. Graph Theory 46 (2004), 15–24.
[4] F. C. Holroyd, and M. Skoviera, Colouring of cubic graphs by Steiner triplesystems, J. Combin. Theory Ser. B. 91 (2004), 57–66.
[5] D. Kral’, E. Macajova, A. Por, and J.-S. Sereni, Characterization results forSteiner triple systems and their application to edge-colorings of cubic graphs, inpreparation.
[6] D. Pal, and M. Skoviera, Colouring cubic graphs by small Steiner triple systems,to appear in Graphs Combin.
[7] V. G. Vizing: Colouring the vertices of a graph with prescribed colours (inRussian), Metody Diskretnogo Analiza Teorii Kodov i Skhem 29 (1976), 3–10.
D. Král’ et al. / Electronic Notes in Discrete Mathematics 29 (2007) 23–27 27
