On multiset colorings of generalized corona graphs
A vertex $k$-coloring of a graph $G$ is a \emph{multiset $k$-coloring} if $M(u)\neq M(v)$ for every edge $uv\in E(G)$, where $M(u)$ and $M(v)$ denote the multisets of colors of the neighbors of $u$ and $v$, respectively. The minimum $k$ for which $G$ has a multiset $k$-coloring is the \emph{multiset...
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doaj-b843fed1e89e48479f0265ce4b24c21b2020-11-25T00:37:47ZengInstitute of Mathematics of the Czech Academy of ScienceMathematica Bohemica0862-79592464-71362016-12-01141443145510.21136/MB.2016.0053-14MB.2016.0053-14On multiset colorings of generalized corona graphsYun FengWensong LinA vertex $k$-coloring of a graph $G$ is a \emph{multiset $k$-coloring} if $M(u)\neq M(v)$ for every edge $uv\in E(G)$, where $M(u)$ and $M(v)$ denote the multisets of colors of the neighbors of $u$ and $v$, respectively. The minimum $k$ for which $G$ has a multiset $k$-coloring is the \emph{multiset chromatic number} $\chi_m(G)$ of $G$. For an integer $\ell\geq0$, the $\ell$-\emph{corona} of a graph $G$, ${\rm cor}^{\ell}(G)$, is the graph obtained from $G$ by adding, for each vertex $v$ in $G$, $\ell$ new neighbors which are end-vertices. In this paper, the multiset chromatic numbers are determined for \mbox{$\ell$-\emph{coronas}} of all complete graphs, the regular complete multipartite graphs and the Cartesian product $K_r\square K_2$ of $K_r$ and $K_2$. In addition, we show that the minimum $\ell$ such that $\chi_m({\rm cor}^{\ell}(G))=2$ never exceeds $\chi(G)-2$, where $G$ is a regular graph and $\chi(G)$ is the chromatic number of $G$.http://mb.math.cas.cz/full/141/4/mb141_4_2.pdf multiset coloring multiset chromatic number generalized corona of a graph neighbor-distinguishing coloring |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Yun Feng Wensong Lin |
spellingShingle |
Yun Feng Wensong Lin On multiset colorings of generalized corona graphs Mathematica Bohemica multiset coloring multiset chromatic number generalized corona of a graph neighbor-distinguishing coloring |
author_facet |
Yun Feng Wensong Lin |
author_sort |
Yun Feng |
title |
On multiset colorings of generalized corona graphs |
title_short |
On multiset colorings of generalized corona graphs |
title_full |
On multiset colorings of generalized corona graphs |
title_fullStr |
On multiset colorings of generalized corona graphs |
title_full_unstemmed |
On multiset colorings of generalized corona graphs |
title_sort |
on multiset colorings of generalized corona graphs |
publisher |
Institute of Mathematics of the Czech Academy of Science |
series |
Mathematica Bohemica |
issn |
0862-7959 2464-7136 |
publishDate |
2016-12-01 |
description |
A vertex $k$-coloring of a graph $G$ is a \emph{multiset $k$-coloring} if $M(u)\neq M(v)$ for every edge $uv\in E(G)$, where $M(u)$ and $M(v)$ denote the multisets of colors of the neighbors of $u$ and $v$, respectively. The minimum $k$ for which $G$ has a multiset $k$-coloring is the \emph{multiset chromatic number} $\chi_m(G)$ of $G$. For an integer $\ell\geq0$, the $\ell$-\emph{corona} of a graph $G$, ${\rm cor}^{\ell}(G)$, is the graph obtained from $G$ by adding, for each vertex $v$ in $G$, $\ell$ new neighbors which are end-vertices. In this paper, the multiset chromatic numbers are determined for \mbox{$\ell$-\emph{coronas}} of all complete graphs, the regular complete multipartite graphs and the Cartesian product $K_r\square K_2$ of $K_r$ and $K_2$. In addition, we show that the minimum $\ell$ such that $\chi_m({\rm cor}^{\ell}(G))=2$ never exceeds $\chi(G)-2$, where $G$ is a regular graph and $\chi(G)$ is the chromatic number of $G$. |
topic |
multiset coloring multiset chromatic number generalized corona of a graph neighbor-distinguishing coloring |
url |
http://mb.math.cas.cz/full/141/4/mb141_4_2.pdf |
work_keys_str_mv |
AT yunfeng onmultisetcoloringsofgeneralizedcoronagraphs AT wensonglin onmultisetcoloringsofgeneralizedcoronagraphs |
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