Packing chromatic number, $(1,1,2,2)$-colorings, and characterizing the Petersen graph
| Autor: | Brešar, Boštjan, Klavžar, Sandi, Rall, Douglas F., Wash, Kirsti |
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| Rok vydání: | 2016 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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 $\Pi_1,\ldots,\Pi_k$, where $\Pi_i$, $i\in [k]$, is an $i$-packing. The following conjecture is posed and studied: if $G$ is a subcubic graph, then $\chi_{\rho}(S(G))\le 5$, where $S(G)$ is the subdivision of $G$. The conjecture is proved for all generalized prisms of cycles. To get this result it is proved that if $G$ is a generalized prism of a cycle, then $G$ is $(1,1,2,2)$-colorable if and only if $G$ is not the Petersen graph. The validity of the conjecture is further proved for graphs that can be obtained from generalized prisms in such a way that one of the two $n$-cycles in the edge set of a generalized prism is replaced by a union of cycles among which at most one is a 5-cycle. The packing chromatic number of graphs obtained by subdividing each of its edges a fixed number of times is also considered. Comment: 16 pages, 4 figures |
| Databáze: | arXiv |
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