Choosability and Edge Choosability of Planar Graphs without Intersecting Triangles

Abstract

Previous article Next article Choosability and Edge Choosability of Planar Graphs without Intersecting TrianglesWei-Fan Wang and Ko-Wei LihWei-Fan Wang and Ko-Wei Lihhttps://doi.org/10.1137/S0895480100376253PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractLet G be a planar graph without two triangles sharing a common vertex. We prove that (1) G is 4-choosable and (2) G is edge-$(\Delta(G)+1)$-choosable when its maximum degree $\Delta(G)\ne 5$.[1] V. Aksionov and , L. Mel'nikov, Some counterexamples associated with the three‐color problem, J. Combin. Theory Ser. B, 28 (1980), 1–9 81d:05027 CrossrefISIGoogle Scholar[2] N. Alon and , M. Tarsi, Colorings and orientations of graphs, Combinatorica, 12 (1992), 125–134 93h:05067 CrossrefISIGoogle Scholar[3] O. Borodin, A generalization of Kotzig's theorem and prescribed edge coloring of planar graphs, Mat. Zametki, 48 (1990), 0–022–28, 160 92e:05046 Google Scholar[4] Oleg Borodin, An extension of Kotzig's theorem on the minimum weight of edges in 3‐polytopes, Math. Slovaca, 42 (1992), 385–389 94a:52024 Google Scholar[5] Paul Erdös, , Arthur Rubin and , Herbert Taylor, Choosability in graphs, Congress. Numer., XXVI, Utilitas Math., Winnipeg, Man., 1980, 125–157 82f:05038 Google Scholar[6] Shai Gutner, The complexity of planar graph choosability, Discrete Math., 159 (1996), 119–130 10.1016/0012-365X(95)00104-5 97f:05182 CrossrefISIGoogle Scholar[7] Roland Häggkvist and , Jeannette Janssen, New bounds on the list‐chromatic index of the complete graph and other simple graphs, Combin. Probab. Comput., 6 (1997), 295–313 10.1017/S0963548397002927 98i:05076 CrossrefGoogle Scholar[8] Google Scholar[9] Ivan Havel, On a conjecture of B. Grünbaum, J. Combinatorial Theory, 7 (1969), 184–186 39:6785 CrossrefGoogle Scholar[10] Martin Juvan, , Bojan Mohar and , Riste Škrekovski, Graphs of degree 4 are 5‐edge‐choosable, J. Graph Theory, 32 (1999), 250–264 10.1002/(SICI)1097-0118(199911)32:3<250::AID-JGT5>3.3.CO;2-I 2000k:05115 CrossrefISIGoogle Scholar[11] A. Kostochka, List edge chromatic number of graphs with large girth, Discrete Math., 101 (1992), 189–201, Special volume to mark the centennial of Julius Petersen's "Die Theorie der regulären Graphs", Part II 10.1016/0012-365X(92)90602-C 93e:05033 CrossrefISIGoogle Scholar[12] Peter Lam, , Baogang Xu and , Jiazhuang Liu, The 4‐choosability of plane graphs without 4‐cycles, J. Combin. Theory Ser. B, 76 (1999), 117–126 10.1006/jctb.1998.1893 2000b:05058 CrossrefISIGoogle Scholar[13] Peter Lam, , Wai Shiu and , Baogang Xu, On structure of some plane graphs with application to choosability, J. Combin. Theory Ser. B, 82 (2001), 285–296 10.1006/jctb.2001.2038 2002d:05105 CrossrefISIGoogle Scholar[14] Maryam Mirzakhani, A small non‐4‐choosable planar graph, Bull. Inst. Combin. Appl., 17 (1996), 15–18 96m:05088 Google Scholar[15] Carsten Thomassen, Every planar graph is 5‐choosable, J. Combin. Theory Ser. B, 62 (1994), 180–181 10.1006/jctb.1994.1062 95f:05045 CrossrefISIGoogle Scholar[16] Carsten Thomassen, 3‐list‐coloring planar graphs of girth 5, J. Combin. Theory Ser. B, 64 (1995), 101–107 10.1006/jctb.1995.1027 96c:05070 CrossrefISIGoogle Scholar[17] V. Vizing, Coloring the vertices of a graph in prescribed colors, Diskret. Analiz, (1976), 0–03–10, 101 58:16371 Google Scholar[18] Margit Voigt, List colourings of planar graphs, Discrete Math., 120 (1993), 215–219 10.1016/0012-365X(93)90579-I 94d:05061 CrossrefISIGoogle Scholar[19] Margit Voigt, A not 3‐choosable planar graph without 3‐cycles, Discrete Math., 146 (1995), 325–328 10.1016/0012-365X(94)00180-9 96g:05119 CrossrefISIGoogle Scholar[20] M. Voigt and , B. Wirth, On 3‐colorable non‐4‐choosable planar graphs, J. Graph Theory, 24 (1997), 233–235 10.1002/(SICI)1097-0118(199703)24:3<233::AID-JGT4>3.3.CO;2-C 97i:05049 CrossrefISIGoogle Scholar[21] Weifan Wang and , Ko‐Wei Lih, The 4‐choosability of planar graphs without 6‐cycles, Australas. J. Combin., 24 (2001), 157–164 1852816 Google Scholar[22] Weifan Wang and , Ko‐Wei Lih, Choosability and edge choosability of planar graphs without five cycles, Appl. Math. Lett., 15 (2002), 561–565 10.1016/S0893-9659(02)80007-6 1889505 CrossrefISIGoogle ScholarKeywordschoosabilityedge choosabilityplanar graph Previous article Next article FiguresRelatedReferencesCited ByDetails On Sufficient Conditions for Planar Graphs to be 5-FlexibleGraphs and Combinatorics, Vol. 38, No. 3 | 14 March 2022 Cross Ref Cover and variable degeneracyDiscrete Mathematics, Vol. 345, No. 4 | 1 Apr 2022 Cross Ref On (3, r)-Choosability of Some Planar GraphsBulletin of the Malaysian Mathematical Sciences Society, Vol. 45, No. 2 | 12 January 2022 Cross Ref Flexibility of planar graphs—Sharpening the tools to get lists of size fourDiscrete Applied Mathematics, Vol. 306 | 1 Jan 2022 Cross Ref DP-4-coloring of planar graphs with some restrictions on cyclesDiscrete Mathematics, Vol. 344, No. 11 | 1 Nov 2021 Cross Ref DP-coloring on planar graphs without given adjacent short cyclesDiscrete Mathematics, Algorithms and Applications, Vol. 13, No. 02 | 10 October 2020 Cross Ref Planar Graphs Without Pairwise Adjacent $3$-, $4$-, $5$-, and $6$-cycle are $4$-choosableTaiwanese Journal of Mathematics, Vol. -1, No. -1 | 1 Jan 2021 Cross Ref Planar graphs without 7-cycles and butterflies are DP-4-colorableDiscrete Mathematics, Vol. 343, No. 8 | 1 Aug 2020 Cross Ref DP-4-colorability of planar graphs without adjacent cycles of given lengthDiscrete Applied Mathematics, Vol. 277 | 1 Apr 2020 Cross Ref List coloring and diagonal coloring for plane graphs of diameter twoApplied Mathematics and Computation, Vol. 363 | 1 Dec 2019 Cross Ref Every planar graph without 4-cycles adjacent to two triangles is DP-4-colorableDiscrete Mathematics, Vol. 342, No. 3 | 1 Mar 2019 Cross Ref Facial Colorings of Plane GraphsJournal of Interconnection Networks, Vol. 19, No. 01 | 28 April 2019 Cross Ref Social Structure Decomposition With Security IssueIEEE Access, Vol. 7 | 1 Jan 2019 Cross Ref Planar graphs without chordal 6-cycles are 4-choosableDiscrete Applied Mathematics, Vol. 244 | 1 Jul 2018 Cross Ref Planar graphs without intersecting 5 -cycles are 4 -choosableDiscrete Mathematics, Vol. 340, No. 8 | 1 Aug 2017 Cross Ref A sufficient condition for a planar graph to be 4 -choosableDiscrete Applied Mathematics, Vol. 224 | 1 Jun 2017 Cross Ref Planar graphs without 4-cycles adjacent to triangles are 4-choosableDiscrete Mathematics, Vol. 339, No. 12 | 1 Dec 2016 Cross Ref List Edge Coloring of Planar Graphs Without Non-Induced 6-CyclesGraphs and Combinatorics, Vol. 31, No. 4 | 12 April 2014 Cross Ref Sufficient conditions for a planar graph to be list edge Δ -colorable and list totally (Δ+1) -colorableDiscrete Mathematics, Vol. 313, No. 5 | 1 Mar 2013 Cross Ref Improper Choosability of Planar Graphs without 4-CyclesYingqian Wang and Lingji XuSIAM Journal on Discrete Mathematics, Vol. 27, No. 4 | 5 December 2013AbstractPDF (192 KB)On Coloring ProblemsHandbook of Combinatorial Optimization | 26 July 2013 Cross Ref Edge choosability and total choosability of planar graphs with no 3-cycles adjacent 4-cyclesDiscrete Mathematics, Vol. 311, No. 20 | 1 Oct 2011 Cross Ref A note on edge-choosability of planar graphs without intersecting 4-cyclesJournal of Applied Mathematics and Computing, Vol. 36, No. 1-2 | 30 May 2010 Cross Ref Edge choosability of planar graphs without 5-cycles with a chordDiscrete Mathematics, Vol. 309, No. 8 | 1 Apr 2009 Cross Ref On 3-choosable planar graphs of girth at least 4Discrete Mathematics, Vol. 309, No. 8 | 1 Apr 2009 Cross Ref Edge-choosability of planar graphs without non-induced 5-cyclesInformation Processing Letters, Vol. 109, No. 7 | 1 Mar 2009 Cross Ref Choosability of toroidal graphs without short cyclesJournal of Graph Theory, Vol. 28 | 1 Jan 2009 Cross Ref Edge-choosability of planar graphs without adjacent triangles or without 7-cyclesDiscrete Mathematics, Vol. 309, No. 1 | 1 Jan 2009 Cross Ref On 3-choosability of planar graphs without certain cyclesInformation Processing Letters, Vol. 107, No. 3-4 | 1 Jul 2008 Cross Ref The 4-choosability of toroidal graphs without intersecting trianglesInformation Processing Letters, Vol. 102, No. 2-3 | 1 Apr 2007 Cross Ref On the sizes of graphs embeddable in surfaces of nonnegative Euler characteristic and their applications to edge choosabilityEuropean Journal of Combinatorics, Vol. 28, No. 1 | 1 Jan 2007 Cross Ref Every toroidal graph without adjacent triangles is (4,1)* -choosableDiscrete Applied Mathematics, Vol. 155, No. 1 | 1 Jan 2007 Cross Ref Bordeaux 3-color conjecture and 3-choosabilityDiscrete Mathematics, Vol. 306, No. 6 | 1 Apr 2006 Cross Ref On 3-choosability of plane graphs without 6-, 7- and 9-cyclesApplied Mathematics-A Journal of Chinese Universities, Vol. 19, No. 1 | 1 Mar 2004 Cross Ref Volume 15, Issue 4| 2002SIAM Journal on Discrete Mathematics435-575 History Published online:01 August 2006 InformationCopyright © 2002 Society for Industrial and Applied MathematicsKeywordschoosabilityedge choosabilityplanar graphMSC codes05C15PDF Download Article & Publication DataArticle DOI:10.1137/S0895480100376253Article page range:pp. 538-545ISSN (print):0895-4801ISSN (online):1095-7146Publisher:Society for Industrial and Applied Mathematics