scholarly journals Fractional Coloring of Triangle-Free Planar Graphs

10.37236/4135 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Zdeněk Dvořák ◽  
Jean-Sébastien Sereni ◽  
Jan Volec
Keyword(s):  
Chromatic Number ◽  
Planar Graphs ◽  
Free Graph ◽  
Fractional Chromatic Number ◽  
Fractional Coloring

We prove that every planar triangle-free graph on $n$ vertices has fractional chromatic number at most $3-3/(3n+1)$.

scholarly journals Determination of Fractional Chromatic Numbers in the Operation of Adding Two Different Graphs

2022 ◽  
Vol 18 (2) ◽  
pp. 161-168
Author(s):  
Junianto Sesa ◽  
Siswanto Siswanto
Keyword(s):  
Graph Theory ◽  
Chromatic Number ◽  
Index Theory ◽  
Chromatic Numbers ◽  
Fractional Chromatic Number ◽  
Path Graph ◽  
Fractional Coloring ◽  
Different Color ◽  
Cycle Graph

The development of graph theory has provided many new pieces of knowledge, one of them is graph color. Where the application is spread in various fields such as the coding index theory. Fractional coloring is multiple coloring at points with different colors where the adjoining point has a different color. The operation in the graph is known as the sum operation. Point coloring can be applied to graphs where the result of operations is from several special graphs.  In this case, the graph summation results of the path graph and the cycle graph will produce the same fractional chromatic number as the sum of the fractional chromatic numbers of each graph before it is operated.

scholarly journals Bipartite Induced Density in Triangle-Free Graphs

10.37236/8650 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Wouter Cames van Batenburg ◽  
Rémi De Joannis de Verclos ◽  
Ross J. Kang ◽  
François Pirot
Keyword(s):  
Chromatic Number ◽  
Minimum Degree ◽  
Free Graph ◽  
Induced Subgraph ◽  
Logarithmic Factor ◽  
Fractional Chromatic Number ◽  
List Chromatic Number

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 $d^2/(2n)$. This is sharp up to a logarithmic factor in $n$. Relatedly, we show that the fractional chromatic number of any such triangle-free graph is at most the minimum of $n/d$ and $(2+o(1))\sqrt{n/\log n}$ as $n\to\infty$. This is sharp up to constant factors. Similarly, we show that the list chromatic number of any such triangle-free graph is at most $O(\min\{\sqrt{n},(n\log n)/d\})$ as $n\to\infty$. Relatedly, we also make two conjectures. First, any triangle-free graph on $n$ vertices has fractional chromatic number at most $(\sqrt{2}+o(1))\sqrt{n/\log n}$ as $n\to\infty$. Second, any  triangle-free graph on $n$ vertices has list chromatic number at most $O(\sqrt{n/\log n})$ as $n\to\infty$.

Girth and fractional chromatic number of planar graphs

2002 ◽  
Vol 39 (3) ◽  
pp. 201-217 ◽  
Author(s):  
Amir Pirnazar ◽  
Daniel H. Ullman
Keyword(s):  
Chromatic Number ◽  
Planar Graphs ◽  
Fractional Chromatic Number

r-Hued coloring of planar graphs without short cycles

2020 ◽  
Vol 12 (03) ◽  
pp. 2050034
Author(s):  
Yuehua Bu ◽  
Xiaofang Wang
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Minimum Integer

A [Formula: see text]-hued coloring of a graph [Formula: see text] is a proper [Formula: see text]-coloring [Formula: see text] such that [Formula: see text] for any vertex [Formula: see text]. The [Formula: see text]-hued chromatic number of [Formula: see text], written [Formula: see text], is the minimum integer [Formula: see text] such that [Formula: see text] has a [Formula: see text]-hued coloring. In this paper, we show that [Formula: see text] if [Formula: see text] and [Formula: see text] is a planar graph without [Formula: see text]-cycles or if [Formula: see text] is a planar graph without [Formula: see text]-cycles and no [Formula: see text]-cycle is intersect with [Formula: see text]-cycles, [Formula: see text], then [Formula: see text], where [Formula: see text].

scholarly journals Another Infinite Sequence of Dense Triangle-Free Graphs

10.37236/1381 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Stephan Brandt ◽  
Tomaž Pisanski
Keyword(s):  
Structural Properties ◽  
Chromatic Number ◽  
Infinite Sequence ◽  
Minimum Degree ◽  
Free Graph ◽  
The Core ◽  
Graph Homomorphisms ◽  
Natural Way

The core is the unique homorphically minimal subgraph of a graph. A triangle-free graph with minimum degree $\delta > n/3$ is called dense. It was observed by many authors that dense triangle-free graphs share strong structural properties and that the natural way to describe the structure of these graphs is in terms of graph homomorphisms. One infinite sequence of cores of dense maximal triangle-free graphs was known. All graphs in this sequence are 3-colourable. Only two additional cores with chromatic number 4 were known. We show that the additional graphs are the initial terms of a second infinite sequence of cores.

scholarly journals Cubical coloring — fractional covering by cuts and semidefinite programming

2015 ◽  
Vol Vol. 17 no.2 (Graph Theory) ◽  
Author(s):  
Robert Šámal
Keyword(s):  
Semidefinite Programming ◽  
Chromatic Number ◽  
Fractional Chromatic Number ◽  
Maximum Cut ◽  
Continuous Mappings ◽  
Polynomial Time Approximation Algorithm ◽  
Eigenvalue Bound ◽  
International Audience ◽  
Graph Parameters ◽  
Fractional Covering

International audience We introduce a new graph parameter that measures fractional covering of a graph by cuts. Besides being interesting in its own right, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with chromatic number, bipartite density, and other graph parameters. We find the value of our parameter for a family of graphs based on hypercubes. These graphs play for our parameter the role that cliques play for the chromatic number and Kneser graphs for the fractional chromatic number. The fact that the defined parameter attains on these graphs the correct value suggests that our definition is a natural one. In the proof we use the eigenvalue bound for maximum cut and a recent result of Engström, Färnqvist, Jonsson, and Thapper [An approximability-related parameter on graphs – properties and applications, DMTCS vol. 17:1, 2015, 33–66]. We also provide a polynomial time approximation algorithm based on semidefinite programming and in particular on vector chromatic number (defined by Karger, Motwani and Sudan [Approximate graph coloring by semidefinite programming, J. ACM 45 (1998), no. 2, 246–265]).

An improved upper bound for the acyclic chromatic number of 1-planar graphs

2020 ◽  
Vol 283 ◽  
pp. 275-291
Author(s):  
Wanshun Yang ◽  
Weifan Wang ◽  
Yiqiao Wang
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Planar Graphs
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