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### Chromatic capacities of graphs and hypergraphs

Joshua Evan Greene
2004 Discrete Mathematics
We prove that if G is a 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.  ...

### The minimum number of edges in a 4-critical graph that is bipartite plus 3 edges

A.V. Kostochka, B.M. Reiniger
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.  ...

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

Tomasz Łuczak, Miklós Simonovits
2008 Discrete Mathematics
Let (G) denote the minimum 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.  ...

### On dispersability of some products of cycles [article]

Samuel S. Joslin, Paul C. Kainen, Shannon Overbay
2021 arXiv   pre-print
We show that the matching book thickness of the Cartesian product of two odd-length cycle-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.  ...

### High-Girth Graphs Avoiding a Minor are Nearly Bipartite

Anna Galluccio, Luis A. Goddyn, Pavol Hell
2001 Journal of combinatorial theory. Series B (Print)
Equivalently, such 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.  ...

### 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.  ...

### Detour Chromatic Numbers

Frank Bullock, Marietjie Frick
2001 Discussiones Mathematicae Graph Theory
The nth detour 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  ...

### 1-subdivisions, fractional chromatic number and Hall ratio [article]

Zdeněk Dvořák and Patrice Ossona de Mendez and Hehui Wu
2020 arXiv   pre-print
c > 0, every 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.  ...

### 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  ...

### Separation Choosability and Dense Bipartite Induced Subgraphs

Louis Esperet, Ross Kang, Stéphan Thomassé
2019 Combinatorics, probability & computing
We show for 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  ...

### Coloring dense graphs via VC-dimension [article]

Tomasz Łuczak, Stéphan Thomassé
2010 arXiv   pre-print
The 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.  ...

### Bipartite induced density in triangle-free graphs [article]

Wouter Cames van Batenburg, Rémi de Joannis de Verclos, Ross J. Kang, François Pirot
2020 arXiv   pre-print
We prove that any triangle-free 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.  ...

### Bipartite Induced Density in Triangle-Free Graphs

Wouter Cames van Batenburg, Rémi De Joannis de Verclos, Ross J. Kang, François Pirot
2020 Electronic Journal of Combinatorics
We prove that any triangle-free 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.  ...