A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2016; you can also visit the original URL.
The file type is application/pdf.
Filters
Proof Pearl: A Probabilistic Proof for the Girth-Chromatic Number Theorem
[chapter]
2012
Lecture Notes in Computer Science
Preserved Fulltext
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)). ...
doi:10.1007/978-3-642-32347-8_27
fatcat:mhfexq2shnblnkg34zhumcfybm
Three coloring via triangle counting
[article]
2022
arXiv
pre-print
Preserved Fulltext
A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2022; you can also visit the original URL.
The file type is application/pdf.
Other Versions
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. ...
arXiv:2203.08136v2
fatcat:cbsx7ps5jjdipencsbgs2ucsmq
MINIMAX DEGREES OF QUASIPLANAR GRAPHS WITH NO SHORT CYCLES OTHER THAN TRIANGLES
2008
Taiwanese journal of mathematics
Preserved Fulltext
A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2012; you can also visit the original URL.
The file type is application/pdf.
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. ...
doi:10.11650/twjm/1500404982
fatcat:t74n574o6rgu7ojhecmrrgusrq
Page 8490 of Mathematical Reviews Vol. , Issue 2004k
[page]
2004
Mathematical Reviews
Preserved Fulltext
The Internet Archive has digitized a microfilm copy of this work. It may be possible to borrow a copy for reading.
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
2014
European journal of combinatorics (Print)
Preserved Fulltext
A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2017; you can also visit the original URL.
The file type is application/pdf.
Other Versions
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 [8] . ...
doi:10.1016/j.ejc.2013.05.002
fatcat:5l2zzgiyjbgldjr6wjqfnxobzq
Three-coloring triangle-free planar graphs in linear time
[article]
2013
arXiv
pre-print
Preserved Fulltext
The Internet Archive has a preservation copy of this work in our general collections.
The file type is application/pdf.
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. ...
arXiv:1302.5121v1
fatcat:tm6tjk46bfgv5fjhiv7ph22egm
Author index to volume
2004
Discrete Mathematics
Preserved Fulltext
A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2019; you can also visit the original URL.
The file type is application/pdf.
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 ! ...
doi:10.1016/s0012-365x(04)00345-0
fatcat:6zribkcwoneppcmbtj3xqfm3qy
Locally Checkable Proofs in Distributed Computing
2016
Theory of Computing
Preserved Fulltext
A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2019; you can also visit the original URL.
The file type is application/pdf.
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. ...
doi:10.4086/toc.2016.v012a019
dblp:journals/toc/GoosS16
fatcat:5mjhwq5q65apxg4yxmol33e6gi
A new proof of 3-colorability of Eulerian triangulations
2011
Ars Mathematica Contemporanea
Preserved Fulltext
A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2012; you can also visit the original URL.
The file type is application/pdf.
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. ...
doi:10.26493/1855-3974.193.8e7
fatcat:vvczovlhrveetfd544fv25i52e
List-coloring embedded graphs
[chapter]
2013
Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms
Preserved Fulltext
A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2019; you can also visit the original URL.
The file type is application/pdf.
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. ...
doi:10.1137/1.9781611973105.72
dblp:conf/soda/DvorakK13
fatcat:kc272dhajfajxh4ypvnauzd2ty
List-coloring embedded graphs
[article]
2012
arXiv
pre-print
Preserved Fulltext
The Internet Archive has a preservation copy of this work in our general collections.
The file type is application/pdf.
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. ...
arXiv:1210.7605v1
fatcat:oqz3p5dqu5c23b47mctc7i2s34
M-degrees of quadrangle-free planar graphs
2009
Journal of Graph Theory
Preserved Fulltext
A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2012; you can also visit the original URL.
The file type is application/pdf.
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 ...
doi:10.1002/jgt.20346
fatcat:xhwcvuwdwjfwla25pbzdujiaka
A five-color theorem for graphs on surfaces
1984
Proceedings of the American Mathematical Society
Preserved Fulltext
A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2019; you can also visit the original URL.
The file type is application/pdf.
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 ...
doi:10.1090/s0002-9939-1984-0728376-7
fatcat:7zirv4h7hfg3rhyxmzmjcm4n4i
A Five-Color Theorem for Graphs on Surfaces
1984
Proceedings of the American Mathematical Society
Preserved Fulltext
A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2017; you can also visit the original URL.
The file type is application/pdf.
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 ...
doi:10.2307/2044501
fatcat:pcim5tmdzjg4pa2rl6abtgyhpm
Cycle Cover with Short Cycles
[chapter]
2005
Lecture Notes in Computer Science
Preserved Fulltext
A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2010; you can also visit the original URL.
The file type is application/pdf.
[11] with 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] . ...
doi:10.1007/978-3-540-31856-9_53
fatcat:j2ytvgrelbhpdhy5im4gizwahy
« Previous
Showing results 1 — 15 out of 47,149 results