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Chromatic capacities of graphs and hypergraphs

2004
*
Discrete Mathematics
*

We prove that if G is a

doi:10.1016/j.disc.2003.06.016
fatcat:gl2j5qnf3bb4nmokxramsapg6m
*graph*on n vertices*with**chromatic**number*and*chromatic*capacity cap, then cap ¿ (1 − o(1)) = √ 2n ln , extending a result of Brightwell and Kohayakawa. ... Given a hypergraph H, the*chromatic*capacity cap(H) of H is the largest k for which there exists a k-coloring of the edges of H such that, for every coloring of the vertices of H*with*the edge colors, ... The lower bound in Theorem 2 is most e ective for a*graph**with**large**chromatic**number*relative to its*number*of vertices n. ...##
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The minimum number of edges in a 4-critical graph that is bipartite plus 3 edges

2015
*
European journal of combinatorics (Print)
*

*Nearly*

*bipartite*k-critical Erdős conjectured that (for k ≥ 4) k-critical

*graphs*should not be "

*nearly*

*bipartite*," in the sense that for

*large*n, deleting few edges from a k-critical

*graph*should not make ... The choosability (a.k.a. choice

*number*or list

*chromatic*

*number*) of G , denoted ch(G ), is the minimum k such that G is k-choosable. ...

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On the minimum degree forcing F-free graphs to be (nearly) bipartite

2008
*
Discrete Mathematics
*

Let (G) denote the minimum

doi:10.1016/j.disc.2007.06.047
fatcat:4p7uoymqdjhm5jvm65bll5txie
*number*of edges to be removed from a*graph*G to make it*bipartite*. ... For each 3-*chromatic**graph*F we determine a parameter (F ) such that for each F-free*graph*G on n vertices*with*minimum degree (G) 2n/( (F ) + 2) + o(n) we have (G) = o(n 2 ), while there are F-free*graphs*... Acknowledgments We started to work on dense F-free*graphs*during our visit at Isaac Newton Institute in September 2003; we wish to thank the Institute for its support and hospitality. ...##
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On dispersability of some products of cycles
[article]

2021
*
arXiv
*
pre-print

We show that the matching book thickness of the Cartesian product of two odd-length cycle-

arXiv:2108.11839v1
fatcat:mdk4tl3rerfvfnbl7245bxexji
*graphs*is five if at least one of the cycles has length 3 or 5. ... Unlike*chromatic**number*, which can be as small as 2 and as*large*as the*number*of vertices,*chromatic*index has only two possible values, ∆(G) or 1 + ∆(G), according to whether G is of Vizing class I or ... so the product of*bipartite*dispersable*graphs*is dispersable and the product of a*bipartite*dispersable*graph*and a*nearly*dispersable*graph*is*nearly*dispersable. ...##
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High-Girth Graphs Avoiding a Minor are Nearly Bipartite

2001
*
Journal of combinatorial theory. Series B (Print)
*

Equivalently, such

doi:10.1006/jctb.2000.2009
fatcat:aaoa5dj2abd3larhdh4nvduqi4
*graphs*have homomorphisms into a*large*odd circuit. In particular,*graphs*of high girth and of bounded genus or bounded tree width are \*nearly**bipartite*" in this sense. ... Let H be a xed*graph*. We show that any H-minor free*graph*of high enough girth has a circular-*chromatic**number*arbitrarily close to two. ... C 2k+1 Thus a*graph*is \*nearly**bipartite*" if its circular*chromatic**number*is close to two. ...##
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Page 4657 of Mathematical Reviews Vol. , Issue 2002G
[page]

2002
*
Mathematical Reviews
*

In particular,

*graphs**with*a sufficiently*large*girth and*with*a bounded genus, or bounded tree width, are “*nearly**bipartite*”. ... Let H be a fixed*graph*. It is proved in this paper that any H- minor free*graph*G*with*a sufficiently*large*girth has a circular*chromatic**number*arbitrarily close to 2. ...##
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Detour Chromatic Numbers

2001
*
Discussiones Mathematicae Graph Theory
*

The nth detour

doi:10.7151/dmgt.1150
fatcat:kbudn4r3dfftjh63nmsb36gjn4
*chromatic**number*, χ n (G) of a*graph*G is the minimum*number*of colours required to colour the vertices of G such that no path*with*more than n vertices is monocoloured. ... We also present some sufficient conditions for a*graph*to have nth*chromatic**number*at most 2. ... In Section 3 we show that*graphs**with**large*enough odd girth, as well as*graphs**with*small enough*bipartite*index, have nth*chromatic**number*at most 2, indicating that having nth detour*chromatic**number*...##
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1-subdivisions, fractional chromatic number and Hall ratio
[article]

2020
*
arXiv
*
pre-print

c > 0, every

arXiv:1812.07327v2
fatcat:vqgotw2mvff4hn2scvsbna4pka
*graph*of sufficiently*large*average degree contains as a subgraph the 1-subdivision of a*graph*of fractional*chromatic**number*at least c. * For every d > 0, there exists a*graph*G of average ... The Hall ratio of a*graph*G is the maximum of |V(H)|/alpha(H) over all subgraphs H of G. Clearly, the Hall ratio of a*graph*is a lower bound for the fractional*chromatic**number*. ... Acknowledgments The results on Hall ratio were inspired by discussions taking place during the first Southwestern German Workshop on*Graph*Theory in Karlsruhe. ...##
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Page 1289 of Mathematical Reviews Vol. , Issue 95c
[page]

1995
*
Mathematical Reviews
*

A

*graph*is*nearly**bipartite*if it contains no odd cycle of length at least five. The authors prove that the complements of*nearly**bipartite**graphs*are strongly perfect. Chinh T. ... {For the entire collection see MR 95b:6801 1.} 95c:05060 05C15 Ravindra, G. (6-RCE-SC; Mysore); Basavayya, D. (6-RCE-SC; Mysore) A characterization of*nearly**bipartite**graphs**with*strongly perfect complements ...##
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Separation Choosability and Dense Bipartite Induced Subgraphs

2019
*
Combinatorics, probability & computing
*

We show for

doi:10.1017/s0963548319000026
fatcat:yjnunsgmvvcstlci77s3dou4hy
*bipartite**graphs*that separation choosability increases*with*(the logarithm of) the minimum degree. This strengthens results of Molloy and Thron and, partially, of Alon. ... For example, does every triangle-free*graph*of minimum degree d contain a*bipartite*induced subgraph of minimum degree Ω(log d) as d→∞? ... settled Conjecture 1.5 (and thus confirmed Conjecture 1.6), in that they have established ( log d/ log log d)*bipartite*induced minimum degree in K r -free*graphs*for every fixed r 3. • Work of Davies ...##
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Coloring dense graphs via VC-dimension
[article]

2010
*
arXiv
*
pre-print

The

arXiv:1007.1670v1
fatcat:q4m3klqqivcgfogfwnmeb35qvu
*large**chromatic**number*follows from the Borsuk-Ulam Theorem. ... In other words, one can find H-free*graphs**with*unbounded*chromatic**number*and minimum degree arbitrarily close to n/3. These H-free*graphs*are derived from a construction of Hajnal. ... It is based on Kneser*graphs*, which confer the*large**chromatic**number*, to which is added a*bipartite**graph*of*large*size, in order to raise the minimum degree. ...##
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Bipartite induced density in triangle-free graphs
[article]

2020
*
arXiv
*
pre-print

We prove that any triangle-free

arXiv:1808.02512v3
fatcat:lvzw7566tbgjnhmdt26lmubgty
*graph*on n vertices*with*minimum degree at least d contains a*bipartite*induced subgraph of minimum degree at least d^2/(2n). ... Second, any triangle-free*graph*on n vertices has list*chromatic**number*at most O(√(n/log n)) as n→∞. ... Acknowledgements We thank Ewan Davies for his insightful remark about regular*graphs*in Conjectures 4.3 and 4.4. ...##
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Bipartite Induced Density in Triangle-Free Graphs

2020
*
Electronic Journal of Combinatorics
*

We prove that any triangle-free

doi:10.37236/8650
fatcat:gkhy6dapx5c7hh4r7ql5hqekum
*graph*on $n$ vertices*with*minimum degree at least $d$ contains a*bipartite*induced subgraph of minimum degree at least $d^2/(2n)$. ... Second, any triangle-free*graph*on $n$ vertices has list*chromatic**number*at most $O(\sqrt{n/\log n})$ as $n\to\infty$. ... Acknowledgements We thank Ewan Davies for his insightful remark about regular*graphs*in Conjectures 4.3 and 4.4. ...##
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Fruit Salad

1997
*
Electronic Journal of Combinatorics
*

Conversations

doi:10.37236/1293
fatcat:rf7qaum7dvbjvnbwyf56pz4dom
*with*Ralph Faudree, Dick Schelp, Bjarne Toft about some of these subjects are also appreciated. The careful reading of the referee improved the presentation. ... 'A problem*with*Hajnal: if each odd cycle of a*graph*G spans a subgraph*with**chromatic**number*at most r then the*chromatic**number*of the*graph*is bounded by a function of r.' ... Property P1 alone easily implies that the*chromatic**number*of*nearly**bipartite**graphs*is at most five. In fact, K 5 shows that P1 alone does not imply more. ...##
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Critical graphs without triangles: An optimum density construction

2013
*
Combinatorica
*

We construct dense, triangle-free,

doi:10.1007/s00493-013-2440-1
fatcat:4yvm5nn55rdgjm3souj75cklia
*chromatic*-critical*graphs*of*chromatic**number*k for all k ≥ 4. ... We also demonstrate (nonconstructively) the existence of dense pentagonand-triangle-free k-critical*graphs*for any k ≥ 4, again*with*a best possible density of > ( 1 4 − ε)n 2 edges for k ≥ 6. ... triangle-free*graphs**with*minimum degree δ ≥ (1 − ε) n 3 and arbitrarily*large**chromatic**number*. ...
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