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

2012
*
Lecture Notes in Computer Science
*

The Girth-Chromatic number theorem is a theorem from

doi:10.1007/978-3-642-32347-8_27
fatcat:mhfexq2shnblnkg34zhumcfybm
*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)). ...##
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Three coloring via triangle counting
[article]

2022
*
arXiv
*
pre-print

In the first partial result toward the Steinberg's now-disproved three

arXiv:2203.08136v2
fatcat:cbsx7ps5jjdipencsbgs2ucsmq
*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*. ...##
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MINIMAX DEGREES OF QUASIPLANAR GRAPHS WITH NO SHORT CYCLES OTHER THAN TRIANGLES

2008
*
Taiwanese journal of mathematics
*

We show that every planar

doi:10.11650/twjm/1500404982
fatcat:t74n574o6rgu7ojhecmrrgusrq
*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. ...##
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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*). ...##
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Short proofs of coloring theorems on planar graphs

2014
*
European journal of combinatorics (Print)
*

In this paper we use the same

doi:10.1016/j.ejc.2013.05.002
fatcat:5l2zzgiyjbgldjr6wjqfnxobzq
*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 [8] . ...##
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Three-coloring triangle-free planar graphs in linear time
[article]

2013
*
arXiv
*
pre-print

Grotzsch's theorem states that every triangle-free planar

arXiv:1302.5121v1
fatcat:tm6tjk46bfgv5fjhiv7ph22egm
*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*. ...##
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Author index to volume

2004
*
Discrete Mathematics
*

Tuza, On

doi:10.1016/s0012-365x(04)00345-0
fatcat:6zribkcwoneppcmbtj3xqfm3qy
*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 ! ...##
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Locally Checkable Proofs in Distributed Computing

2016
*
Theory of Computing
*

Among the most difficult

doi:10.4086/toc.2016.v012a019
dblp:journals/toc/GoosS16
fatcat:5mjhwq5q65apxg4yxmol33e6gi
*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. ...##
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A new proof of 3-colorability of Eulerian triangulations

2011
*
Ars Mathematica Contemporanea
*

Using the existence of noncrossing Eulerian circuits in Eulerian plane

doi:10.26493/1855-3974.193.8e7
fatcat:vvczovlhrveetfd544fv25i52e
*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. ...##
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List-coloring embedded graphs
[chapter]

2013
*
Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms
*

This also enables us to find such a

doi:10.1137/1.9781611973105.72
dblp:conf/soda/DvorakK13
fatcat:kc272dhajfajxh4ypvnauzd2ty
*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*. ...##
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List-coloring embedded graphs
[article]

2012
*
arXiv
*
pre-print

This also enables us to find such a

arXiv:1210.7605v1
fatcat:oqz3p5dqu5c23b47mctc7i2s34
*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*. ...##
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M-degrees of quadrangle-free planar graphs

2009
*
Journal of Graph Theory
*

We determine the maximum possible M-degrees for planar, projective-planar and toroidal

doi:10.1002/jgt.20346
fatcat:xhwcvuwdwjfwla25pbzdujiaka
*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 ...##
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A five-color theorem for graphs on surfaces

1984
*
Proceedings of the American Mathematical Society
*

We prove that if a

doi:10.1090/s0002-9939-1984-0728376-7
fatcat:7zirv4h7hfg3rhyxmzmjcm4n4i
*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 ...##
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A Five-Color Theorem for Graphs on Surfaces

1984
*
Proceedings of the American Mathematical Society
*

We prove that if a

doi:10.2307/2044501
fatcat:pcim5tmdzjg4pa2rl6abtgyhpm
*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 ...##
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Cycle Cover with Short Cycles
[chapter]

2005
*
Lecture Notes in Computer Science
*

[11] with

doi:10.1007/978-3-540-31856-9_53
fatcat:j2ytvgrelbhpdhy5im4gizwahy
*cycle*-size*bound*in uniform*graphs*. ... Itai et al. [13] 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 [11] . ...
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