论文标题
同时着色的顶点和图形的发生率
Simultaneous coloring of vertices and incidences of graphs
论文作者
论文摘要
图$ g $的$ n $ -subdivision是通过更换$ g $的长度$ n $的路径而构建的图形，而$ g $的每个边缘和$ g $的$ m $ popper aus $ g $是一个图形，其顶点与$ g $和$ g $相同的两个顶点，最多是$ m $的$ g $的任何两个顶点。图$ g^{\ frac {m} {n}} $是$ g $的$ n $ subdivision的$ m $ - 功率。在[ N. Iradmusa，M。Mozafari-nia，关于$ \ frac {3} {3} {3} $的彩色的注释 - 子四分之图的功率，第1卷。 79，No.3，2021]猜想$ \ frac {3} {3} {3} $的色度数 - 具有最大度$δ\ geq 2 $的图形功率最多是$2δ+1 $。在本文中，我们介绍了顶点和图表的同时着色，并表明同时正确着色的顶点和$ g $的颜色的最小颜色数量，由$χ_{vi}（vi}（g）$表示，等于$ G^{\ FRAC {\ freac freac \ frac {\ frac {\ frac {\ frac {3} $}} $ c}} = 3}。同样，通过确定上述参数的确切值或上限，我们研究了某些类图的猜想的正确性，例如$ k $ degeNerated图，周期，森林，完整的图形和常规的两部分图。此外，我们研究了该新的色数与图形的其他参数之间的关系。
An $n$-subdivision of a graph $G$ is a graph constructed by replacing a path of length $n$ instead of each edge of $G$ and an $m$-power of $G$ is a graph with the same vertices as $G$ and any two vertices of $G$ at distance at most $m$ are adjacent. The graph $G^{\frac{m}{n}}$ is the $m$-power of the $n$-subdivision of $G$. In [M. N. Iradmusa, M. Mozafari-Nia, A note on coloring of $\frac{3}{3}$-power of subquartic graphs, Vol. 79, No.3, 2021] it was conjectured that the chromatic number of $\frac{3}{3}$-power of graphs with maximum degree $Δ\geq 2$ is at most $2Δ+1$. In this paper, we introduce the simultaneous coloring of vertices and incidences of graphs and show that the minimum number of colors for simultaneous proper coloring of vertices and incidences of $G$, denoted by $χ_{vi}(G)$, is equal to the chromatic number of $G^{\frac{3}{3}}$. Also by determining the exact value or the upper bound for the said parameter, we investigate the correctness of the conjecture for some classes of graphs such as $k$-degenerated graphs, cycles, forests, complete graphs, and regular bipartite graphs. In addition, we investigate the relationship between this new chromatic number and the other parameters of graphs.
