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### Proof Pearl: A Probabilistic Proof for the Girth-Chromatic Number Theorem [chapter]

Lars Noschinski
2012 Lecture Notes in Computer Science
The Girth-Chromatic number theorem is a theorem from graph theory, stating that graphs with arbitrarily large girth and chromatic number exist.  ...  graph without cycles will be denoted by ∞.  ...  If for all graphs H both short-cycles(H, k) = ∅ ⇒ P (H, k) and finite(short-cycles(H, k)) ∧ short-cycles(H, k) = ∅ ∧P (H − choose-v(H, k)) ⇒ P (H, k) Lemma 10 ( 10 Lower Bound for χ(G)).  ...

### Three coloring via triangle counting [article]

Zachary Hamaker, Vincent Vatter
2022 arXiv   pre-print
In the first partial result toward the Steinberg's now-disproved three coloring conjecture, Abbott and Zhou used a counting argument to show that every planar graph without cycles of lengths 4 through  ...  We show how this result, combined with Kostochka and Yancey's resolution of Ore's conjecture for k = 4, implies that every planar graph without cycles of lengths 4 through 8 is 3-colorable.  ...  Every planar graph without cycles of lengths 4 through 7 is 3-colorable.  ...

### MINIMAX DEGREES OF QUASIPLANAR GRAPHS WITH NO SHORT CYCLES OTHER THAN TRIANGLES

Oleg V. Borodin, Anna O. Ivanova, Alexandr V. Kostochka, Naeem N. Sheikh
2008 Taiwanese journal of mathematics
We show that every planar graph G without leaves and 4-and 5-cycles has M -degree at most 5, which bound is sharp.  ...  We also show that every planar graph G without leaves and cycles of length from 4 to 7 has M -degree at most 4, which bound is attained even on planar graphs with no cycles of length from 4 to arbitrarily  ...  Corollary 1 Let G be either a projective-planar graph without 4-and 5-cycles, or a toroidal or Kleinian graph without cycles of length from 4 to 6.  ...

### Page 8490 of Mathematical Reviews Vol. , Issue 2004k [page]

2004 Mathematical Reviews
of plane graphs without 6-, 7- and 9-cycles.  ...  Mat. 82 (1957), 76-92; MR 19,876c] proved that the line graph of a snark (a non-edge- 3-colorable cubic graph) is a 4-coloring-snark (a non-edge-4- colorable 4-regular graph).  ...

### Short proofs of coloring theorems on planar graphs

Oleg V. Borodin, Alexandr V. Kostochka, Bernard Lidický, Matthew Yancey
2014 European journal of combinatorics (Print)
In this paper we use the same bound to give short proofs of other known theorems on 3-coloring of planar graphs, among whose is the Gr\"unbaum-Aksenov Theorem that every planar with at most three triangles  ...  A recent lower bound on the number of edges in a k-critical n-vertex graph by Kostochka and Yancey yields a half-page proof of the celebrated Gr\"otzsch Theorem that every planar triangle-free graph is  ...  Erdős suggested relaxing the conjecture and asked for the smallest k such that every planar graph without cycles of length 4 to k is 3-colorable. The best known bound for k is 7  .  ...

### Three-coloring triangle-free planar graphs in linear time [article]

Zdenek Dvorak, Ken-ichi Kawarabayashi, Robin Thomas
2013 arXiv   pre-print
Grotzsch's theorem states that every triangle-free planar graph is 3-colorable. Several relatively simple proofs of this fact were provided by Thomassen and other authors.  ...  We design a linear-time algorithm to find a 3-coloring of a given triangle-free planar graph. The algorithm avoids using any complex data structures, which makes it easy to implement.  ...  Short proof of Grötzsch's theorem Let G be a plane graph.  ...

### Author index to volume

2004 Discrete Mathematics
Tuza, On short cycles through prescribed vertices of a graph (1-2) 67-74 G . orlich, A., M. Pil! sniak, M. Wo! zniak and I.A. Zio"o, A note on embedding graphs without short cycles (1-2) 75-77 G !  ...  nski, Which non-regular bipartite integral graphs with maximum degree four do not have 71 as eigenvalues? (1-2) 15-24 Boland, J., F. Buckley and M.  ...  Tuza, On short cycles through prescribed vertices of a graph (1-2) 67-74 G . orlich, A., M. Pil! sniak, M. Wo! zniak and I.A. Zio"o, A note on embedding graphs without short cycles (1-2) 75-77 G !  ...

### Locally Checkable Proofs in Distributed Computing

Mika Göös, Jukka Suomela
2016 Theory of Computing
Among the most difficult graph properties are proving that a graph is symmetric (has a non-trivial automorphism), which requires Ω(n 2 ) bits per node, and proving that a graph is not 3-colorable, which  ...  For example, it is easy to prove that a graph is bipartite: the locally checkable proof gives a 2-coloring of the graph, which only takes 1 bit per node.  ...  For example, in the case of non-bipartiteness, each short cycle has an odd number of nodes, but the long cycle is composed of an even number of short cycles, and is therefore a no-instance.  ...

### A new proof of 3-colorability of Eulerian triangulations

Mu-Tsun Tsai, Douglas B. West
2011 Ars Mathematica Contemporanea
Using the existence of noncrossing Eulerian circuits in Eulerian plane graphs, we give a short constructive proof of the theorem of Heawood that Eulerian triangulations are 3-colorable.  ...  Figure 1 : 1 A non-3-colorable Eulerian plane graph  ...  Our proof uses the fact that Eulerian plane graphs have "non-crossing" Eulerian circuits.  ...

### List-coloring embedded graphs [chapter]

Zdeněk Dvořák, Ken-ichi Kawarabayashi
2013 Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms
This also enables us to find such a coloring when it exists. The idea of the algorithm can be applied to other similar problems, e.g., 5-list-coloring of graphs on surfaces. *  ...  For any fixed surface Σ of genus g, we give an algorithm to decide whether a graph G of girth at least five embedded in Σ is colorable from an assignment of lists of size three in time O(|V (G)|).  ...  The algorithm A standard dynamic programming approach enables us to deal with short non-F -contractible cycles.  ...

### List-coloring embedded graphs [article]

Zdenek Dvorak, Ken-ichi Kawarabayashi
2012 arXiv   pre-print
This also enables us to find such a coloring when it exists. The idea of the algorithm can be applied to other similar problems, e.g., 5-list-coloring of graphs on surfaces.  ...  For any fixed surface Sigma of genus g, we give an algorithm to decide whether a graph G of girth at least five embedded in Sigma is colorable from an assignment of lists of size three in time O(|V(G)|  ...  The algorithm A standard dynamic programming approach enables us to deal with short non-F -contractible cycles.  ...

### M-degrees of quadrangle-free planar graphs

Oleg V. Borodin, Alexandr V. Kostochka, Naeem N. Sheikh, Gexin Yu
2009 Journal of Graph Theory
We determine the maximum possible M-degrees for planar, projective-planar and toroidal graphs without leaves and 4-cycles. In particular, for planar and projective-planar graphs this maximum is 7.  ...  In order to get upper bounds on the game chromatic number, He et al showed that every planar graph G without leaves and 4cycles has M-degree at most 8 and gave an example of such a graph with M-degree  ...  Corollary 1 . 1 If G is a planar graph without 4-cycles, then (i) G has an edge-partition into a forest and a subgraph H with (H) ≤ 6; (ii) the game chromatic number and the game coloring number of G is  ...

### A five-color theorem for graphs on surfaces

Joan P. Hutchinson
1984 Proceedings of the American Mathematical Society
We prove that if a graph embeds on a surface with all edges suitably short, then the vertices of the graph can be five-colored.  ...  The motivation is that a graph embedded with short edges is locally a planar graph and hence should not require many more than four colors.  ...  A cycle in an embedded graph is said to be null-homologous or non-null-homologous if it is an nc-cycle whose removal does or does not, respectively, disconnect the graph; we abbreviate the latter by calling  ...

### A Five-Color Theorem for Graphs on Surfaces

Joan P. Hutchinson
1984 Proceedings of the American Mathematical Society
We prove that if a graph embeds on a surface with all edges suitably short, then the vertices of the graph can be five-colored.  ...  The motivation is that a graph embedded with short edges is locally a planar graph and hence should not require many more than four colors.  ...  A cycle in an embedded graph is said to be null-homologous or non-null-homologous if it is an nc-cycle whose removal does or does not, respectively, disconnect the graph; we abbreviate the latter by calling  ...

### Cycle Cover with Short Cycles [chapter]

Nicole Immorlica, Mohammad Mahdian, Vahab S. Mirrokni
2005 Lecture Notes in Computer Science
 with cycle-size bound in uniform graphs.  ...  Itai et al.  proved an upper bound on the length of such a cycle cover in ¾-connected graphs and gave an algorithm to find it.  ...  Cycle cover with simple short cycles Our techniques also give results on the bounded cycle cover problem  .  ...
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