An injective k-coloring of a graph G is a mapping $f:V(G)\to \{1,2,\dots ,k\}$ such that for any two vertices $v_1,v_2\in V(G)$, $f(v_1)\ne f(v_2)$ if $N(v_1)\cap N(v_2)\ne \varnothing$. The injective chromatic number of a graph G, denoted by ${\chi }_i(G)$, is the smallest integer k such that G has an injective k-coloring. In this paper, we prove that for a Halin graph G, ${\chi }_i(G)\le \mathrm{\Delta }(G)+2$. Moreover, ${\chi }_i(G)\le \mathrm{\Delta }(G)+1$ if $\mathrm{\Delta }(G)\ge 6$. Also, we show that for a triangle-free planar graph G without intersecting 4-cycles, ${\chi }_i(G)\le \mathrm{\Delta }(G)+6$ if $\mathrm{\Delta }(G)\ge 20$.

中文翻译：

---

平面图的内射着色

图G的内射k着色是映射$F：V（G）\to \{\mathrm{1个}，2，\dots ，ķ\}$ 这样对于任何两个顶点 $v_{\mathrm{1个}}，v_2\in V（G）$， $F（v_{\mathrm{1个}}）\ne F（v_2）$ 如果 $ñ（v_{\mathrm{1个}}）\cap ñ（v_2）\ne \varnothing$。图G的内射色数，表示为${\chi }_{\mathrm{一世}}（G）$是最小整数k，使得G具有内射性k着色。在本文中，我们证明对于Halin图G，${\chi }_{\mathrm{一世}}（G）\le \mathrm{\Delta }（G）+2$。此外，${\chi }_{\mathrm{一世}}（G）\le \mathrm{\Delta }（G）+\mathrm{1个}$ 如果 $\mathrm{\Delta }（G）\ge 6$。此外，我们表明，对于不具有4个周期相交的无三角形平面图G，${\chi }_{\mathrm{一世}}（G）\le \mathrm{\Delta }（G）+6$ 如果 $\mathrm{\Delta }（G）\ge 20$。