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On list 3-dynamic coloring of near-triangulations

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 نشر من قبل Yulai Ma
 تاريخ النشر 2019
  مجال البحث
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An $r$-dynamic $k$-coloring of a graph $G$ is a proper $k$-coloring such that for any vertex $v$, there are at least $min{r, deg_G(v) }$ distinct colors in $N_G(v)$. The $r$-dynamic chromatic number $chi_r^d(G)$ of a graph $G$ is the least $k$ such that there exists an $r$-dynamic $k$-coloring of $G$. The list $r$-dynamic chromatic number of a graph $G$ is denoted by $ch_r^d(G)$. Loeb et al. $[11]$ showed that $ch_3^d(G)leq 10$ for every planar graph $G$, and there is a planar graph $G$ with $chi_3^d(G)= 7$. In this paper, we study a special class of planar graphs which have better upper bounds of $ch_3^d(G)$. We prove that $ch_3^d(G) leq 6$ if $G$ is a planar graph which is near-triangulation, where a near-triangulation is a planar graph whose bounded faces are all 3-cycles.



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