4-Critical Graphs on Surfaces Without Contractible $(\le\!4)$-Cycles
Abstract
We show that if $G$ is a 4-critical graph embedded in a fixed surface $\Sigma$ so that every contractible cycle has length at least 5, then $G$ can be expressed as $G=G'\cup G_1\cup G_2\cup\cdots\cup G_k$, where $|V(G')|$ and $k$ are bounded by a constant (depending linearly on the genus of $\Sigma$) and $G_1, \ldots, G_k$ are graphs (of unbounded size) whose structure we describe exactly. The proof is computer assisted---we use a computer to enumerate all plane 4-critical graphs of girth 5 with a precolored cycle of length at most 16 that are used in the basic case of the inductive proof of the statement.
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
V. A. Aksenov, On continuation of 3-colouring of planar graphs, Diskret. Anal. Novosibirsk, 26 (1974), pp. 3--19 (in Russian).
2.
O. V. Borodin, A. V. Kostochka, B. Lidický, and M. Yancey, Short proofs of coloring theorems on planar graphs, European J. Combin., 36 (2014), pp. 314--321.
3.
O. Borodin, Z. Dvořák, A. Kostochka, B. Lidický, and M. Yancey, Planar 4-critical graphs with four triangles,, European J. Combin., to appear.
4.
Z. Dvořák and K. Kawarabayashi, Choosability of Planar Graphs of Girth 5, J. Combin., submitted.
5.
Z. Dvořák, D. Král', and R. Thomas, Three-Coloring Triangle-Free Graphs on Surfaces III. Graphs of Girth Five, J. Combin., submitted.
6.
J. Gimbel and C. Thomassen, Coloring graphs with fixed genus and girth, Trans. Amer. Math. Soc., 349 (1997), pp. 4555--4564.
7.
H. Grötzsch, Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe, 8 (1959), pp. 109--120.
8.
L. Postle, $3$-List-Coloring Graphs of Girth at Least Five on Surfaces, manuscript.
9.
C. Thomassen, Grötzsch's 3-color theorem and its counterparts for the torus and the projective plane, J. Combin. Theory Ser. B, 62 (1994), pp. 268--279.
10.
C. Thomassen, The chromatic number of a graph of girth 5 on a fixed surface, J. Combin. Theory Ser. B, 87 (2003), pp. 38--71.
11.
R. Thomas and B. Walls, Three-coloring Klein bottle graphs of girth five, J. Combin. Theory Ser. B, 92 (2004), pp. 115--135.
12.
B. Walls, Coloring Girth Restricted Graphs on Surfaces, Ph.D. thesis, Georgia Institute of Technology, Atlanta, GA, 1999.
13.
D. Youngs, 4-chromatic projective graphs, J. Combin. Theory Ser. B, 21 (1996), pp. 219--227.
Information & Authors
Information
Published In
Copyright
© 2014, Society for Industrial and Applied Mathematics.
History
Submitted: 14 May 2013
Accepted: 7 January 2014
Published online: 25 March 2014
Keywords
MSC codes
Authors
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- Three-coloring triangle-free graphs on surfaces III. Graphs of girth fiveJournal of Combinatorial Theory, Series B, Vol. 145 | 1 Nov 2020
- 3-coloring triangle-free planar graphs with a precolored 9-cycleEuropean Journal of Combinatorics, Vol. 68 | 1 Feb 2018
- Fine Structure of 4-Critical Triangle-Free Graphs I. Planar Graphs with Two Triangles and 3-Colorability of ChainsSIAM Journal on Discrete Mathematics, Vol. 32, No. 3 | 24 July 2018
- Fine Structure of 4-Critical Triangle-Free Graphs II. Planar Triangle-Free Graphs with Two Precolored 4-CyclesSIAM Journal on Discrete Mathematics, Vol. 31, No. 2 | 11 May 2017
- 3‐Coloring Triangle‐Free Planar Graphs with a Precolored 8‐CycleJournal of Graph Theory, Vol. 80, No. 2 | 3 December 2014
- 3-Coloring Triangle-Free Planar Graphs with a Precolored 9-CycleCombinatorial Algorithms | 7 June 2015
View Options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].