This paper presents infinitely many new examples of triangle-free uniquely 3-edge colorable cubic graphs. The only such graph previously known was given by Tutte in 1976

Belcastro, Sarah-Marie

Haas, Ruth

English

arXiv

Smith College: Smith ScholarWorks

Masthead Logo Smith ScholarWorksMathematics and Statistics: Faculty Publications Mathematics and Statistics2015Triangle-free Uniquely 3-Edge Colorable CubicGraphsSarah-Marie BelcastroUniversity of Massachusetts Amherst, smbelcas@toroidalsnark.netRuth HaasSmith College, rhaas@smith.eduFollow this and additional works at: https://scholarworks.smith.edu/mth_facpubsPart of the Mathematics CommonsThis Article has been accepted for inclusion in Mathematics and Statistics: Faculty Publications by an authorized administrator of Smith ScholarWorks.For more information, please contact scholarworks@smith.eduRecommended CitationBelcastro, Sarah-Marie and Haas, Ruth, "Triangle-free Uniquely 3-Edge Colorable Cubic Graphs" (2015). Mathematics and Statistics:Faculty Publications, Smith College, Northampton, MA.https://scholarworks.smith.edu/mth_facpubs/5arXiv:1508.06934v1  [math.CO]  27 Aug 2015TRIANGLE-FREE UNIQUELY 3-EDGE COLORABLECUBIC GRAPHSSARAH-MARIE BELCASTRO AND RUTH HAASAbstract. This paper presents infinitely many new examples of tri-angle-free uniquely 3-edge colorable cubic graphs. The only such graphpreviously known was given by Tutte in 1976.1. HistoryRecall that a cubic graph is 3-regular, that a proper 3-edge coloring assignscolors to edges such that no two incident edges receive the same color, thatedge-Kempe chains are maximal sequences of edges that alternate betweentwo colors, and that a Hamilton cycle includes all vertices of a graph.It is well known that a cubic graph with a Hamilton cycle is 3-edge col-orable, as the Hamilton cycle is even (and thus 2-edge colorable) and itscomplement is a matching (that can be monochromatically colored). Auniquely 3-edge colorable cubic graph must have exactly three Hamiltoncycles, each an edge-Kempe chain in one of the(32)pairs of colors. Theconverse is not true, as a cubic graph may have some colorings with Hamil-ton edge-Kempe chains and other colorings with non-Hamilton edge-Kempechains; examples are given in [12].The literature classifying uniquely 3-edge colorable cubic graphs is sparse;there is no complete characterization [7]. It is well known that the propertyof being uniquely 3-edge colorable is invariant under application of ∆ − Ytransformations. It was conjectured that every simple planar cubic graphwith exactly three Hamilton cycles contains a triangle [13, Cantoni], and alsothat every simple planar uniquely 3-edge colorable cubic graph contains atriangle [3]. The latter conjecture is proved in [4], where it is also shownthat if a simple planar cubic graph has exactly three Hamilton cycles, thenit contains a triangle if and only if it is uniquely 3-edge colorable.Tutte, in a 1976 paper about the average number of Hamilton cycles in agraph [13], offhandedly remarks that one example of a nonplanar triangle-free uniquely 3-edge colorable cubic graph is the generalized Petersen graphP (9, 2), pictured in Figure 1. He describes the graph as two 9-cycles a0 . . . a8,b0 . . . b8, with additional edges aib2i and index arithmetic done modulo 9.2010 Mathematics Subject Classification. 05C10, 05C15.Key words and phrases. edge coloring, unique coloring, triangle-free, cubic graphs.Ruth Haas’s work partially supported by NSF grant DMS-1143716 and Simons Foun-dation Award Number 281291.12 SARAH-MARIE BELCASTRO AND RUTH HAAS012345678012345678Figure 1. The generalized Petersen graph P (9, 2) labeledwith Tutte’s indices.(The generalized Petersen graph P (m, 2) is defined analogously, and in factthe known cubic graphs with exactly three Hamilton cycles and multipledistinct 3-edge colorings are P (6k + 3, 2) for k > 1 [12].) It appears thatthe search for examples of triangle-free nonplanar uniquely 3-edge colorablecubic graphs ended with Tutte, or at least that any further efforts have beenunsuccessful. Multiple sources ([6], [7], [9]) note that Tutte’s example is theonly known triangle-free nonplanar example. It has been conjectured [3]that P (9, 2) is the only example. In Section 2 we give infinitely many suchgraphs.2. New examples of triangle-free nonplanar uniquely 3-edgecolorable cubic graphsIn [2] the authors introduced the following construction. Consider two cu-bic graphs G1, G2, and form G1 G2 by choosing a vertex vi in Gi (i = 1, 2),removing vi from Gi (i = 1, 2), and adding a matching of three edges joiningthe three neighbors of v1 with the three neighbors of v2. Of course there aremany ways to choose v1, v2, and many ways to identify their incident edges,so the construction is not unique. However, it is reversible; given a cubicgraph G with a 3-edge cut, we may decompose G = G1 G2. In that paperwe proved the following.Theorem 2.1. [3.8 of [2]] Let G1, G2 be cubic graphs and ai the number of3-edge colorings of Gi. Then G1 G2 has a1a2 edge colorings.Define G to be the infinite family of graphs consisting of all graphs ofthe form G G · · · G. This leads to the following corollaries of Theorem2.1.Theorem 2.2. If G is a uniquely 3-edge colorable graph, then all graphs inG are uniquely 3-edge colorable.TRIANGLE-FREE UNIQUELY 3-EDGE COLORABLE CUBIC GRAPHS 3Figure 2. A nonplanar, triangle-free, uniquely 3-edge col-orable graph with 34 vertices.Figure 3. A nonplanar, triangle-free, uniquely 3-edge col-orable graph with 34 vertices that is nonisomorphic to thatshown in Figure 2.Proof. The proof proceeds by induction on the number of copies of G.Corollary 2.3. All members of the infinite family P (9, 2)are uniquely 3-edge colorable.Note.In [5], Goldwasser and Zhang proved that if a uniquely 3-edge colorablegraph has an edge cut of size 3 or 4 such that each remaining component con-tains a cycle, then the graph can be decomposed into two smaller uniquely3-edge colorable graphs. It seems they did not observe the reverse construc-tion.2.1. Examples and Properties. The smallest member of P (9, 2) is ofcourse P (9, 2), which has 18 vertices. For every integer k > 1 there aremultiple graphs in P (9, 2) with 16k + 2 vertices. Nonisomorphic exampleswith k = 2 are shown in Figures 2 and 3.The graphs in P (9, 2) are clearly all nonplanar. We show next that thereare graphs in P (9, 2) of every other orientable and nonorientable genus.4 SARAH-MARIE BELCASTRO AND RUTH HAASFigure 4. Embeddings of P (9, 2) on the torus (left) andprojective plane (right)Theorem 2.4. Every graph in P (9, 2) with 16k+2 vertices has orientableand nonorientable genus at most k. Further, there is a large subfamily ofgraphs in P (9, 2) , each of which has 16k + 2 vertices and orientable andnonorientable genus exactly k.Proof. We will show by induction that any graph Qk created using the -construction with k copies of P (9, 2) has orientable and nonorientable genusat most k. The base case holds because P (9, 2) embeds on both the torus(see Figure 4 (left)) and on the projective plane (see Figure 4 (right)).Now consider Qk, a graph created using the -construction with k copiesof P (9, 2). Qk was obtained by removing and associating v ∈ P (9, 2) andsome w ∈ Qk−1 via the -construction, where Qk−1 is some graph createdusing k−1 copies of P (9, 2) that has genus k−1 or less by inductive hypoth-esis. Let Q̂k be the graph produced by simply identifying the vertices v andw. The graph Q̂k has two blocks that meet at this vertex and by Theorem1 of [1], the genus of Q̂k is the sum of the genera of the blocks (so it is k).Replacing the cut vertex by a 3-edge cut to implement the constructiondoes not increase the genus, which completes the proof of the upper boundon genus.A copy of a subdivision of K3,3 is highlighted in the embedding of P (9, 2)shown in Figure 5. There are four vertices {t1, t2, t3, t4} whose edges are notinvolved in the subdividedK3,3. Any (or all) of {t1, t2, t3, t4} can be removedand the resulting graph will still have a K3,3 minor. If Qk is formed suchthat in each copy of P (9, 2) only (some) of vertices {t1, t2, t3, t4} are used inthe construction then there will still be k disjoint copies of subdivisions ofK3,3 in Qk. The genus of a graph is the sum of the genera of its components[1, Cor. 2], so using this construction Qk has a minor with orientable (resp.TRIANGLE-FREE UNIQUELY 3-EDGE COLORABLE CUBIC GRAPHS 5Figure 5. P (9, 2) with a copy of a subdivision of K3,3 highlighted.nonorientable) genus exactly k. It is straightforward to draw an embeddingof sample Qk on a surface of orientable or nonorientable genus k.3. ConclusionWhile we have provided infinitely many examples of triangle-free nonpla-nar uniquely 3-edge colorable cubic graphs, it is still unknown whether otherexamples exist. All our examples support Zhang’s conjecture [14] that ev-ery triangle-free uniquely 3-edge colorable cubic graph contains a Petersengraph minor. That conjecture remains open.References1. J. Battle, F. Harary, Y. Kodama, and J.W.T. Youngs, Additivity of the Genus of aGraph, Bull. Amer. Math. Soc. 68 (1962), 565–568.2. s.-m. belcastro and R. Haas, Counting edge-Kempe-equivalence classes for 3-edge-colored cubic graphs, with R. Haas, Discrete Mathematics, 325 (2014), 77–84.3. S. Fiorini and R.J. Wilson, Edge colourings of graphs, in Selected topics in graphtheory, ed. L. Beineke and R. Wilson, Academic Press, 1978, 103–126.4. T.G. Fowler, Unique Coloring of Planar Graphs, dissertation,Georgia Institute of Technology, November, 1998; available atpeople.math.gatech.edu/~thomas/FC/fowlerphd.pdf.5. J.L. Goldwasser and C.-Q. Zhang, On minimal counterexamples to a conjecture aboutunique edge-3-coloring, Festschrift for C. St. J. A. Nash-Williams, Congr. Numer. 113(1996), 143–152.6. J.L. Goldwasser and C.-Q. Zhang, Uniquely Edge-3-Colorable Graphs and Snarks,Graphs and Combinatorics 16 (2000), 257–267.7. J. Gross and J. Yellen, Handbook of Graph Theory, CRC Press, Boca Raton, 2004.8. R. Isaacs, Infinite families of nontrivial trivalent graphs which are not Tait colorable,Amer. Math. Monthly 82 (1975), 221–239.9. T. Jensen and B. Toft, Graph Coloring Problems, Wiley Interscience Series, New York,1995.10. B. Mohar and C. Thomassen, Graphs on Surfaces, Johns Hopkins University Press,Baltimore and London, 2001.11. N. Robertson, D. Sanders, P. D. Seymour and R. Thomas, The four colour theorem,J. Combin. Theory Ser. B. 70 (1997), 2–44.6 SARAH-MARIE BELCASTRO AND RUTH HAAS12. A. Thomason, Cubic graphs with three hamiltonian cycles are not always uniquely edgecolorable, J. Graph Theory 6 (1982), 219–221.13. W.T. Tutte, Hamiltonian circuits, Colloquio Internazional sulle Teorie Combinatorics,Atti dei Convegni Lincei. 17 (1976), Accad. Naz. Lincei, Roma I, 193–199.14. C.-Q. Zhang, Hamiltonian weights and unique edge-3-colorings of cubic graphs, J. ofGraph Theory 20 (1995), 91–99.Department of Mathematics and Statistics, Smith College, Northampton,MA 01063 USAE-mail address: smbelcas@toroidalsnark.netDepartment of Mathematics and Statistics, Smith College, Northampton,MA 01063 USAE-mail address: rhaas@smith.edu

Triangle-free Uniquely 3-Edge Colorable Cubic Graphs

This paper presents infinitely many new examples of triangle-free uniquely 3-edge colorable cubic graphs.  The only such graph previously known was given by Tutte in 1976

belcastro, sarah-marie

University of Calgary Journal Hosting

Contributions to Discrete Mathematics (E-Journal, University of Calgary)

Volume 10, Number 2, Pages 39–44ISSN 1715-0868TRIANGLE-FREE UNIQUELY 3-EDGE COLORABLECUBIC GRAPHSSARAH-MARIE BELCASTRO AND RUTH HAASAbstract. This paper presents infinitely many new examples of tri-angle-free uniquely 3-edge colorable cubic graphs. The only such graphpreviously known was given by Tutte in 1976.1. HistoryRecall that a cubic graph is 3-regular, that a proper 3-edge coloring assignscolors to edges such that no two incident edges receive the same color, thatedge-Kempe chains are maximal sequences of edges that alternate betweentwo colors, and that a Hamilton cycle includes all vertices of a graph.It is well known that a cubic graph with a Hamilton cycle is 3-edge col-orable, as the Hamilton cycle is even (and thus 2-edge colorable) and itscomplement is a matching (that can be monochromatically colored). Auniquely 3-edge colorable cubic graph must have exactly three Hamiltoncycles, each an edge-Kempe chain in one of the(32)pairs of colors. Theconverse is not true, as a cubic graph may have some colorings with Hamil-ton edge-Kempe chains and other colorings with non-Hamilton edge-Kempechains; examples are given in [12].The literature classifying uniquely 3-edge colorable cubic graphs is sparse;there is no complete characterization [7]. It is well known that the propertyof being uniquely 3-edge colorable is invariant under application of ∆ − Ytransformations. It was conjectured that every simple planar cubic graphwith exactly three Hamilton cycles contains a triangle [13, Cantoni], and alsothat every simple planar uniquely 3-edge colorable cubic graph contains atriangle [3]. The latter conjecture is proved in [4], where it is also shownthat if a simple planar cubic graph has exactly three Hamilton cycles, thenit contains a triangle if and only if it is uniquely 3-edge colorable.Tutte, in a 1976 paper about the average number of Hamilton cycles in agraph [13], offhandedly remarks that one example of a nonplanar triangle-free uniquely 3-edge colorable cubic graph is the generalized Petersen graphReceived by the editors April 7, 2015, and in revised form July 1, 2015.2010 Mathematics Subject Classification. 05C10, 05C15.Key words and phrases. edge coloring, unique coloring, triangle-free, cubic graphs.Ruth Haas’s work partially supported by NSF grant DMS-1143716 and Simons Foun-dation Award Number 281291.c©2015 University of Calgary3940 SARAH-MARIE BELCASTRO AND RUTH HAAS012345678012345678Figure 1. The generalized Petersen graph P (9, 2) labeledwith Tutte’s indices.P (9, 2), pictured in Figure 1. He describes the graph as two 9-cycles a0 . . . a8,b0 . . . b8, with additional edges aib2i and index arithmetic done modulo 9.The generalized Petersen graph P (m, 2) is defined analogously, and in factthe known cubic graphs with exactly three Hamilton cycles and multipledistinct 3-edge colorings are P (6k + 3, 2) for k > 1 [12]. It appears thatthe search for examples of triangle-free nonplanar uniquely 3-edge colorablecubic graphs ended with Tutte, or at least that any further efforts have beenunsuccessful. Multiple sources ([6], [7], [9]) note that Tutte’s example is theonly known triangle-free nonplanar example. It has been conjectured [3]that P (9, 2) is the only example. In Section 2 we give infinitely many suchgraphs.2. New examples of triangle-free nonplanar uniquely 3-edgecolorable cubic graphsIn [2] the authors introduced the following construction: Consider twocubic graphs, G1 and G2, and form G1 G2 by choosing a vertex vi inGi (i = 1, 2), removing vi from Gi (i = 1, 2), and adding a matching of threeedges joining the three neighbors of v1 with the three neighbors of v2. Ofcourse there are many ways to choose v1, v2, and many ways to identify theirincident edges, so the construction is not unique. However, it is reversible;given a cubic graph G with a 3-edge cut, we may decompose G = G1 G2.In that paper we proved the following result:Theorem 2.1 (3.8 of [2]). Let G1, G2 be cubic graphs and ai the number of3-edge colorings of Gi. Then G1 G2 has a1a2 edge colorings.Define G to be the infinite family of graphs consisting of all graphs ofthe form G G · · · G. This leads to the following corollaries of Theorem2.1:TRIANGLE-FREE UNIQUELY 3-EDGE COLORABLE CUBIC GRAPHS 41Figure 2. A nonplanar, triangle-free, uniquely 3-edge col-orable graph with 34 vertices.Figure 3. A nonplanar, triangle-free, uniquely 3-edge col-orable graph with 34 vertices that is nonisomorphic to thatshown in Figure 2.Theorem 2.2. If G is a uniquely 3-edge colorable graph, then all graphs inG are uniquely 3-edge colorable.Proof. The proof proceeds by induction on the number of copies of G.Corollary 2.3. All members of the infinite family P (9, 2) are uniquely3-edge colorable.Note. In [5], Goldwasser and Zhang proved that if a uniquely 3-edgecolorable graph has an edge cut of size 3 or 4 such that each remainingcomponent contains a cycle, then the graph can be decomposed into twosmaller uniquely 3-edge colorable graphs. It seems they did not observe thereverse construction.2.1. Examples and Properties. The smallest member of P (9, 2) is ofcourse P (9, 2), which has 18 vertices. For every integer k > 1 there aremultiple graphs in P (9, 2) with 16k + 2 vertices. Nonisomorphic exampleswith k = 2 are shown in Figures 2 and 3.The graphs in P (9, 2) are clearly all nonplanar. We show next that thereare graphs in P (9, 2) of every nonzero orientable and nonorientable genus.42 SARAH-MARIE BELCASTRO AND RUTH HAASFigure 4. Embeddings of P (9, 2) on the torus (left) andprojective plane (right)Theorem 2.4. Every graph in P (9, 2) with 16k+ 2 vertices has orientableand nonorientable genus at most k. Further, there is a large subfamily ofgraphs in P (9, 2) , each of which has 16k + 2 vertices and orientable andnonorientable genus exactly k.Proof. We will show by induction that any graph Qk created using the -construction with k copies of P (9, 2) has orientable and nonorientable genusat most k. The base case holds because P (9, 2) embeds on both the torus(see Figure 4 (left)) and on the projective plane (see Figure 4 (right)).Now consider Qk, a graph created using the -construction with k copiesof P (9, 2). The graph Qk was obtained by removing and associating v ∈P (9, 2) and some w ∈ Qk−1 via the -construction, where Qk−1 is somegraph created using k−1 copies of P (9, 2) that has genus k−1 or less by theinductive hypothesis. Let Q̂k be the graph produced by simply identifyingthe vertices v and w. The graph Q̂k has two blocks that meet at this vertex,so by Theorem 1 of [1] the genus of Q̂k is the sum of the genera of the blocks,which is k. Replacing the cut vertex by a 3-edge cut to implement theconstruction does not increase the genus, which completes the proof of theupper bound on genus.A copy of a subdivision of K3,3 is highlighted in the embedding of P (9, 2)shown in Figure 5. There are four vertices {t1, t2, t3, t4} whose edges are notinvolved in the subdivided K3,3. Any (or all) of {t1, t2, t3, t4} can be removedand the resulting graph will still have a K3,3 minor. If Qk is formed suchthat in each copy of P (9, 2) only (some) of vertices {t1, t2, t3, t4} are used inthe construction, then there will still be k disjoint copies of subdivisions ofK3,3 in Qk. The genus of a graph is the sum of the genera of its components[1, Cor. 2], so using this construction Qk has a minor with orientable (resp.TRIANGLE-FREE UNIQUELY 3-EDGE COLORABLE CUBIC GRAPHS 43Figure 5. P (9, 2) with a copy of a subdivision of K3,3 highlighted.nonorientable) genus exactly k. It is straightforward to draw an embeddingof sample Qk on a surface of orientable or nonorientable genus k.3. ConclusionWhile we have provided infinitely many examples of triangle-free nonpla-nar uniquely 3-edge colorable cubic graphs, it is still unknown whether otherexamples exist. All our examples support Zhang’s conjecture [14] that ev-ery triangle-free uniquely 3-edge colorable cubic graph contains a Petersengraph minor. That conjecture remains open.References1. J. Battle, F. Harary, Y. Kodama, and J. W. T. Youngs, Additivity of the genus of agraph, Bull. Amer. Math. Soc. 68 (1962), 565–568. MR 01553132. s.-m. Belcastro and R. Haas, Counting edge-Kempe-equivalence classes for 3-edge-colored cubic graphs, Discrete Math. 325 (2014), 77–84. MR 31812363. S. Fiorini and R. J. Wilson, Edge colorings of graphs, ch. 5, pp. 103–126, AcademicPress, 1978.4. T. Fowler, Unique coloring of planar graphs, ProQuest LLC, Ann Arbor, MI, 1998,Thesis (Ph.D.)–Georgia Institute of Technology. MR 26987145. J. Goldwasser and C.-Q. Zhang, On minimal counterexamples to a conjecture aboutunique edge-3-coloring, Congr. Numer. 113 (1996), 143–152, Festschrift for C. St. J.A. Nash-Williams. MR 13937066. , Uniquely edge-3-colorable graphs and snarks, Graphs Combin. 16 (2000),no. 3, 257–267. MR 17821867. J. Gross and J. Yellen (eds.), Handbook of graph theory, Discrete Mathematics and itsApplications (Boca Raton), CRC Press, Boca Raton, FL, 2004. MR 20351868. R. Isaacs, Infinite families of nontrivial trivalent graphs which are not Tait colorable,Amer. Math. Monthly 82 (1975), 221–239.9. T. Jensen and B. Toft, Graph coloring problems, Wiley-Interscience Series in DiscreteMathematics and Optimization, John Wiley & Sons, Inc., New York, 1995, A Wiley-Interscience Publication. MR 130425410. B. Mohar and C. Thomassen, Graphs on surfaces, Johns Hopkins Studies in the Math-ematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2001. MR 184444911. N. Robertson, D. Sanders, P. Seymour, and R. Thomas, The four-colour theorem, J.Combin. Theory Ser. B 70 (1997), no. 1, 2–44. MR 144125844 SARAH-MARIE BELCASTRO AND RUTH HAAS12. A. Thomason, Cubic graphs with three hamiltonian cycles are not always uniquely edgecolorable, J. of Graph Theory 6 (1982), no. 2, 219–221.13. W. T. Tutte, Hamiltonian circuits, Colloquio Internazional sulle Teorie Combinatorics(Lincei, Roma I), Atti dei Convegni Lincei, vol. 17, Accad. Nazi. dei Lincei, 1976,pp. 91–99.14. C.-Q. Zhang, Hamiltonian weights and unique 3-edge-colorings of cubic graphs, J. ofGraph Theory 20 (1995), no. 1, 91–99.Department of Mathematics and Statistics, Smith College, Northampton,MA 01063 USAE-mail address: smbelcas@toroidalsnark.netDepartment of Mathematics and Statistics, Smith College, Northampton,MA 01063 USAE-mail address: rhaas@smith.edu

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Triangle-free Uniquely 3-Edge Colorable Cubic Graphs

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