Color-blind index in graphs of very low degree
| Autor: | Diemunsch, Jennifer, Graber, Nathan, Kramer, Lucas, Larsen, Victor, Nelsen, Lauren M., Nelsen, Luke L., Sigler, Devon, Stolee, Derrick, Suer, Charlie |
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| Rok vydání: | 2015 |
| Předmět: | |
| Zdroj: | Discrete Applied Mathematics, Volume 225, 10 July 2017, Pages 122-129 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.dam.2017.03.006 |
| Popis: | Let $c:E(G)\to [k]$ be an edge-coloring of a graph $G$, not necessarily proper. For each vertex $v$, let $\bar{c}(v)=(a_1,\ldots,a_k)$, where $a_i$ is the number of edges incident to $v$ with color $i$. Reorder $\bar{c}(v)$ for every $v$ in $G$ in nonincreasing order to obtain $c^*(v)$, the color-blind partition of $v$. When $c^*$ induces a proper vertex coloring, that is, $c^*(u)\neq c^*(v)$ for every edge $uv$ in $G$, we say that $c$ is color-blind distinguishing. The minimum $k$ for which there exists a color-blind distinguishing edge coloring $c:E(G)\to [k]$ is the color-blind index of $G$, denoted $\operatorname{dal}(G)$. We demonstrate that determining the color-blind index is more subtle than previously thought. In particular, determining if $\operatorname{dal}(G) \leq 2$ is NP-complete. We also connect the color-blind index of a regular bipartite graph to 2-colorable regular hypergraphs and characterize when $\operatorname{dal}(G)$ is finite for a class of 3-regular graphs. Comment: 10 pages, 3 figures, and a 4 page appendix |
| Databáze: | arXiv |
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