论文标题
紧密的线性绑定到$（p_5，k_1+（k_1 \ cup k_3）的色度数字）$ - 免费图形
A tight linear bound to the chromatic number of $(P_5, K_1+(K_1\cup K_3))$-free graphs
论文作者
论文摘要
令$ f_1 $和$ f_2 $为两个不相交图。 Union $ f_1 \ cup f_2 $是带顶点套装$ v（f_1）\ cup v（f_2）$ and edge set $ e（f_1）\ cup e（f_2）$的图表\ {xy \; | \; x \ in V（f_1）\ mbox {and} y \ in V（f_2）\} $。在本文中，我们将特征呈现给$（p_5，k_1 \ cup k_3）$ - 免费图形，证明$χ（g）\le2Ω（g）-1 $如果$ g $是$（p_5，k_1 \ cup k_3）$。基于此结果，我们进一步证明$χ（g）\ le $ max $ \ {2Ω（g），15 \} $，如果$ g $是$（p_5，k_1+（k_1 \ cup k_3））$ - 免费图形 - 免费图形，并构建一个无限的家族$（p_5，k_1+$ g_1+cup g_1 $ g_1 $ g_1 $ g_3） $χ（g）=2Ω（g）$。
Let $F_1$ and $F_2$ be two disjoint graphs. The union $F_1\cup F_2$ is a graph with vertex set $V(F_1)\cup V(F_2)$ and edge set $E(F_1)\cup E(F_2)$, and the join $F_1+F_2$ is a graph with vertex set $V(F_1)\cup V(F_2)$ and edge set $E(F_1)\cup E(F_2)\cup \{xy\;|\; x\in V(F_1)\mbox{ and } y\in V(F_2)\}$. In this paper, we present a characterization to $(P_5, K_1\cup K_3)$-free graphs, prove that $χ(G)\le 2ω(G)-1$ if $G$ is $(P_5, K_1\cup K_3)$-free. Based on this result, we further prove that $χ(G)\le $max$\{2ω(G),15\}$ if $G$ is a $(P_5,K_1+( K_1\cup K_3))$-free graph, and construct an infinite family of $(P_5, K_1+( K_1\cup K_3))$-free graphs such that every graph $G$ in the family satisfies $χ(G)=2ω(G)$.
