论文标题
同时着色的顶点和外平面图的发生率
Simultaneous coloring of vertices and incidences of Outerplanar graphs
论文作者
论文摘要
a $ vi $ - 中等的$ k $ - 颜色图$ g $的颜色是所有顶点和图形的着色，在集合中，任何两个相邻或事件元素在集合$ v（g）\ cup i（g）$中的任何两个相邻或事件元素都会获得独特的颜色，其中$ i（g）是$ g $的$ i（g）$。 $ vi $ -simultanus颜色编号，用$χ_{vi}（g）$表示，是最小的整数$ k $，因此$ g $具有$ vi $ $ vi $ -simultane的$ k $ coloring。在[ Mozafari-nia，M。N。Iradmusa，关于$ \ frac {3} {3} {3} $的彩色的注释 - 子四分之图的功率，第1卷。 79，第3期，2021年] $ vi $ -simimultanate的适当颜色具有最高学位$ 4 $的图形，他们推测，对于任何图形$ g $，最高度$δ\ geq 2 $，$ vi $ $ vi $ -simultane的$ G $的颜色最多为$2Δ+1 $。在[ Mozafari-Nia，M。N。Iradmusa，图表的顶点和染色率的同时着色，Arxiv：2205.07189，2022]]猜想的某些图形的猜想正确性，例如$ k $ $ k $ - $ - $ - $ -DEKENERADER图形，cycles，cycles，cycles，cycles，cycles of Cycles，coccles of Crests，forests，forsects thragephs，常规图形，bipartite图形。在本文中，我们证明$ vi $ -simultane的色度数量$ g $是$δ+2 $或$δ+3 $，其中$δ$是$ g $的最高度。
A $vi$-simultaneous proper $k$-coloring of a graph $G$ is a coloring of all vertices and incidences of the graph in which any two adjacent or incident elements in the set $V(G)\cup I(G)$ receive distinct colors, where $I(G)$ is the set of incidences of $G$. The $vi$-simultaneous chromatic number, denoted by $χ_{vi}(G)$, is the smallest integer $k$ such that $G$ has a $vi$-simultaneous proper $k$-coloring. In [M. Mozafari-Nia, M. N. Iradmusa, A note on coloring of $\frac{3}{3}$-power of subquartic graphs, Vol. 79, No.3, 2021] $vi$-simultaneous proper coloring of graphs with maximum degree $4$ is investigated and they conjectured that for any graph $G$ with maximum degree $Δ\geq 2$, $vi$-simultaneous proper coloring of $G$ is at most $2Δ+1$. In [M. Mozafari-Nia, M. N. Iradmusa, Simultaneous coloring of vertices and incidences of graphs, arXiv:2205.07189, 2022] the correctness of the conjecture for some classes of graphs such as $k$-degenerated graphs, cycles, forests, complete graphs, regular bipartite graphs is investigated. In this paper, we prove that the $vi$-simultaneous chromatic number of any outerplanar graph $G$ is either $Δ+2$ or $Δ+3$, where $Δ$ is the maximum degree of $G$.
