For a graph $G$, $\chi(G)$ $(\omega(G))$ denote its chromatic (clique) number. A $P_5$ is the chordless path on five vertices, and a $4$-$wheel$ is the graph consisting of a chordless cycle on four vertices $C_4$ plus an additional vertex adjacent to all the vertices of the $C_4$. In this paper, we show that every ($P_5$, $4$-wheel)-free graph $G$ satisfies $\chi(G)\leq \frac{3}{2}\omega(G)$. Moreover, this bound is almost tight. That is, there is a class of ($P_5$, $4$-wheel)-free graphs $\cal L$ such that every graph $H\in \cal L$ satisfies $\chi(H)\geq\frac{10}{7}\omega(H)$. This generalize/improve several previously known results in the literature.

翻译：对于一个G$G$, $\chi( G) $( g) $( g) $) 表示它的色( clotic) 号。 A$_ 5美元是5个脊椎上的无弦路径, 4美元轮值为4美元, 由4个脊椎上的无弦循环构成的图表, 4美元加上与所有脊椎相邻的另外1个顶点。 在本文中, 我们显示每张( P_ 5美元, 4美元轮式) 无色( g) $( g)\ leq\ leq\ frac\ { 3 ⁇ 2 ⁇ omega( G) $( g) 。 此外, 这个约束几乎是紧凑的。 也就是说, 每张( P_ 5美元, 4美元) 轮式无弦的图表有1 美元/ call L$( ), 每张H\ $\ cle$\ clex lex $\ chile ( h)\ g)\ geqq\\\\\\\\\\\\\\ 7\ \\ omega( h) 7\ omga( H) $) $( H) $。 这个概括/ =7\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\