Packing chromatic vertex-critical graphs
The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $V_i$, $i\in [k]$, where vertices in $V_i$ are pairwise at distance at least $i+1$. Packing chromatic vertex-critical graphs, $\chi_{\rho}$-critical f...
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Discrete Mathematics & Theoretical Computer Science
2019-02-01
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Series: | Discrete Mathematics & Theoretical Computer Science |
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Online Access: | https://dmtcs.episciences.org/4878/pdf |
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author | Sandi Klavžar Douglas F. Rall |
author_facet | Sandi Klavžar Douglas F. Rall |
author_sort | Sandi Klavžar |
collection | DOAJ |
description | The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest
integer $k$ such that the vertex set of $G$ can be partitioned into sets $V_i$,
$i\in [k]$, where vertices in $V_i$ are pairwise at distance at least $i+1$.
Packing chromatic vertex-critical graphs, $\chi_{\rho}$-critical for short, are
introduced as the graphs $G$ for which $\chi_{\rho}(G-x) < \chi_{\rho}(G)$
holds for every vertex $x$ of $G$. If $\chi_{\rho}(G) = k$, then $G$ is
$k$-$\chi_{\rho}$-critical. It is shown that if $G$ is $\chi_{\rho}$-critical,
then the set $\{\chi_{\rho}(G) - \chi_{\rho}(G-x):\ x\in V(G)\}$ can be almost
arbitrary. The $3$-$\chi_{\rho}$-critical graphs are characterized, and
$4$-$\chi_{\rho}$-critical graphs are characterized in the case when they
contain a cycle of length at least $5$ which is not congruent to $0$ modulo
$4$. It is shown that for every integer $k\ge 2$ there exists a
$k$-$\chi_{\rho}$-critical tree and that a $k$-$\chi_{\rho}$-critical
caterpillar exists if and only if $k\le 7$. Cartesian products are also
considered and in particular it is proved that if $G$ and $H$ are
vertex-transitive graphs and ${\rm diam(G)} + {\rm diam}(H) \le
\chi_{\rho}(G)$, then $G\,\square\, H$ is $\chi_{\rho}$-critical. |
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format | Article |
id | doaj.art-b28f75f16b484e539bfb1466dc7463e2 |
institution | Directory Open Access Journal |
issn | 1365-8050 |
language | English |
last_indexed | 2024-04-25T01:57:44Z |
publishDate | 2019-02-01 |
publisher | Discrete Mathematics & Theoretical Computer Science |
record_format | Article |
series | Discrete Mathematics & Theoretical Computer Science |
spelling | doaj.art-b28f75f16b484e539bfb1466dc7463e22024-03-07T15:39:17ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502019-02-01Vol. 21 no. 3Graph Theory10.23638/DMTCS-21-3-84878Packing chromatic vertex-critical graphsSandi KlavžarDouglas F. RallThe packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $V_i$, $i\in [k]$, where vertices in $V_i$ are pairwise at distance at least $i+1$. Packing chromatic vertex-critical graphs, $\chi_{\rho}$-critical for short, are introduced as the graphs $G$ for which $\chi_{\rho}(G-x) < \chi_{\rho}(G)$ holds for every vertex $x$ of $G$. If $\chi_{\rho}(G) = k$, then $G$ is $k$-$\chi_{\rho}$-critical. It is shown that if $G$ is $\chi_{\rho}$-critical, then the set $\{\chi_{\rho}(G) - \chi_{\rho}(G-x):\ x\in V(G)\}$ can be almost arbitrary. The $3$-$\chi_{\rho}$-critical graphs are characterized, and $4$-$\chi_{\rho}$-critical graphs are characterized in the case when they contain a cycle of length at least $5$ which is not congruent to $0$ modulo $4$. It is shown that for every integer $k\ge 2$ there exists a $k$-$\chi_{\rho}$-critical tree and that a $k$-$\chi_{\rho}$-critical caterpillar exists if and only if $k\le 7$. Cartesian products are also considered and in particular it is proved that if $G$ and $H$ are vertex-transitive graphs and ${\rm diam(G)} + {\rm diam}(H) \le \chi_{\rho}(G)$, then $G\,\square\, H$ is $\chi_{\rho}$-critical.https://dmtcs.episciences.org/4878/pdfmathematics - combinatorics |
spellingShingle | Sandi Klavžar Douglas F. Rall Packing chromatic vertex-critical graphs Discrete Mathematics & Theoretical Computer Science mathematics - combinatorics |
title | Packing chromatic vertex-critical graphs |
title_full | Packing chromatic vertex-critical graphs |
title_fullStr | Packing chromatic vertex-critical graphs |
title_full_unstemmed | Packing chromatic vertex-critical graphs |
title_short | Packing chromatic vertex-critical graphs |
title_sort | packing chromatic vertex critical graphs |
topic | mathematics - combinatorics |
url | https://dmtcs.episciences.org/4878/pdf |
work_keys_str_mv | AT sandiklavzar packingchromaticvertexcriticalgraphs AT douglasfrall packingchromaticvertexcriticalgraphs |