The List-Chromatic Number of Complete Multipartite Hypergraphs and Multiple Covers by Independent Sets

2020 ◽  
Vol 107 (3-4) ◽  
pp. 499-508
Author(s):  
D. A. Shabanov ◽  
T. M. Shaikheeva
Keyword(s):  
Chromatic Number ◽  
Independent Sets ◽  
Complete Multipartite ◽  
List Chromatic Number

scholarly journals Bipartite Graphs whose Squares are not Chromatic-Choosable

10.37236/4343 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Seog-Jin Kim ◽  
Boram Park
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Bipartite Graphs ◽  
Total Coloring ◽  
Natural Question ◽  
List Total Coloring ◽  
Multipartite Graphs ◽  
Complete Multipartite ◽  
List Chromatic Number ◽  
Total Coloring Conjecture

The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. Let $\chi(H)$ and $\chi_{\ell}(H)$ be the chromatic number and the list chromatic number of $H$, respectively. A graph $H$ is called chromatic-choosable if $\chi_{\ell} (H) = \chi(H)$. It is an interesting problem to find graphs that are chromatic-choosable.Motivated by the List Total Coloring Conjecture, Kostochka and Woodall (2001) proposed the List Square Coloring Conjecture which states that $G^2$ is chromatic-choosable for every graph $G$. Recently, Kim and Park showed that the List Square Coloring Conjecture does not hold in general by finding a family of graphs whose squares are complete multipartite graphs and are not chromatic choosable. It is a well-known fact that the List Total Coloring Conjecture is true if the List Square Coloring Conjecture holds for special class of bipartite graphs. Hence a natural question is whether $G^2$ is chromatic-choosable or not for every bipartite graph $G$.In this paper, we give a bipartite graph $G$ such that $\chi_{\ell} (G^2) \neq \chi(G^2)$. Moreover, we show that the value $\chi_{\ell}(G^2) - \chi(G^2)$ can be arbitrarily large.

List Colourings of Regular Hypergraphs

2012 ◽  
Vol 21 (1-2) ◽  
pp. 315-322 ◽  
Author(s):  
DAVID SAXTON ◽  
ANDREW THOMASON
Keyword(s):  
Chromatic Number ◽  
Uniform Hypergraph ◽  
List Chromatic Number

We show that the list chromatic number of a simpled-regularr-uniform hypergraph is at least (1/2rlog(2r2) +o(1)) logdifdis large.

A General Upper Bound on the List Chromatic Number of Locally Sparse Graphs

2002 ◽  
Vol 11 (1) ◽  
pp. 103-111 ◽  
Author(s):  
VAN H. VU
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Present Author ◽  
Upper Bounds ◽  
Constant Factor ◽  
Chromatic Index ◽  
Sparse Graphs ◽  
Critical Graphs ◽  
Multiplicative Constant ◽  
List Chromatic Number

Suppose that G is a graph with maximum degree d(G) such that, for every vertex v in G, the neighbourhood of v contains at most d(G)2/f (f > 1) edges. We show that the list chromatic number of G is at most Kd(G)/log f, for some positive constant K. This result is sharp up to the multiplicative constant K and strengthens previous results by Kim [9], Johansson [7], Alon, Krivelevich and Sudakov [3], and the present author [18]. This also motivates several interesting questions.As an application, we derive several upper bounds for the strong (list) chromatic index of a graph, under various assumptions. These bounds extend earlier results by Faudree, Gyárfás, Schelp and Tuza [6] and Mahdian [13] and determine, up to a constant factor, the strong (list) chromatic index of a random graph. Another application is an extension of a result of Kostochka and Steibitz [10] concerning the structure of list critical graphs.

scholarly journals On θ-commutators and the corresponding non-commuting graphs

2017 ◽  
Vol 15 (1) ◽  
pp. 1530-1538
Author(s):  
S. Shalchi ◽  
A. Erfanian ◽  
M. Farrokhi DG
Keyword(s):  
Chromatic Number ◽  
Independent Sets ◽  
Graphs Of Groups

Abstract The θ-commutators of elements of a group with respect to an automorphism are introduced and their properties are investigated. Also, corresponding to θ-commutators, we define the θ-non-commuting graphs of groups and study their correlations with other notions. Furthermore, we study independent sets in θ-non-commuting graphs, which enable us to evaluate the chromatic number of such graphs.

scholarly journals Colorings of the Graph K ᵐ 2 + Kn

2020 ◽  
pp. 297-305
Author(s):  
Le Xuan Hung
Keyword(s):  
Chromatic Number ◽  
Colorable Graph ◽  
List Chromatic Number

In this paper, we characterize chromatically unique, determine list-chromatic number and characterize uniquely list colorability of the graph G = Km 2 + Kn. We shall prove that G is χ-unique, ch(G) = m + n, G is uniquely 3-list colorable graph if and only if 2m + n > 7 and m > 2

scholarly journals List Coloring Hypergraphs

10.37236/401 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Penny Haxell ◽  
Jacques Verstraete
Keyword(s):  
Chromatic Number ◽  
The Other ◽  
List Coloring ◽  
Other Hand ◽  
List Chromatic Number ◽  
Linear Hypergraphs ◽  
Linear Hypergraph

Let $H$ be a hypergraph and let $L_v : v \in V(H)$ be sets; we refer to these sets as lists and their elements as colors. A list coloring of $H$ is an assignment of a color from $L_v$ to each $v \in V(H)$ in such a way that every edge of $H$ contains a pair of vertices of different colors. The hypergraph $H$ is $k$-list-colorable if it has a list coloring from any collection of lists of size $k$. The list chromatic number of $H$ is the minimum $k$ such that $H$ is $k$-list-colorable. In this paper we prove that every $d$-regular three-uniform linear hypergraph has list chromatic number at least $(\frac{\log d}{5\log \log d})^{1/2}$ provided $d$ is large enough. On the other hand there exist $d$-regular three-uniform linear hypergraphs with list chromatic number at most $\log_3 d+3$. This leaves the question open as to the existence of such hypergraphs with list chromatic number $o(\log d)$ as $d \rightarrow \infty$.

The Strong Perfect Graph Conjecture for Planar Graphs

1973 ◽  
Vol 25 (1) ◽  
pp. 103-114 ◽  
Author(s):  
Alan Tucker
Keyword(s):  
Chromatic Number ◽  
Planar Graphs ◽  
Independent Set ◽  
Independent Sets ◽  
Clique Number ◽  
Perfect Graph ◽  
Stability Number ◽  
Partition Number ◽  
Minimum Number ◽  
The Stability

A graph G is called γ-perfect if ƛ (H) = γ(H) for every vertex-generated subgraph H of G. Here, ƛ(H) is the clique number of H (the size of the largest clique of H) and γ(H) is the chromatic number of H (the minimum number of independent sets of vertices that cover all vertices of H). A graph G is called α-perfect if α(H) = θ(H) for every vertex-generated subgraph H of G, where α (H) is the stability number of H (the size of the largest independent set of H) and θ(H) is the partition number of H (the minimum number of cliques that cover all vertices of H).

Sign in / Sign up

Export Citation Format

Share Document