We present an algorithm to color a graph G with no triangle and no induced 7-vertex path (i.e., a $\{P_7,C_3\}$-free graph), where every vertex is assigned a list of possible colors which is a subset of $\{1,2,3\}$. While this is a special case of the problem solved in Bonomo et al. (2018) [1], that does not require the absence of triangles, the algorithm here is both faster and conceptually simpler. The complexity of the algorithm is $O(|V(G){|}^5(|V(G)|+|E(G)|))$, and if G is bipartite, it improves to $O(|V(G){|}^2(|V(G)|+|E(G)|))$.

Moreover, we prove that there are finitely many minimal obstructions to list 3-coloring $\{P_t,C_3\}$-free graphs if and only if $t\le 7$. This implies the existence of a polynomial time certifying algorithm for list 3-coloring in $\{P_7,C_3\}$-free graphs. We furthermore determine other cases of $t,\ell$, and k such that the family of minimal obstructions to list k-coloring in $\{P_t,C_{\ell }\}$-free graphs is finite.

我们提出了一种算法，可以为没有三角形且没有诱导7顶点路径的图形G着色（即$\{P_7，C_3\}$-无图），其中为每个顶点分配了可能的颜色列表，这些颜色是 $\{\mathrm{1个}，2，3\}$。虽然这是Bonomo等人解决的问题的特例。（2018）[1]，它不需要三角形，这里的算法既更快又概念上更简单。该算法的复杂度为$Ø（|V（G）{|}^5（|V（G）|+|Ë（G）|））$，并且如果G是二分的，它将改善为$Ø（|V（G）{|}^2（|V（G）|+|Ë（G）|））$。

此外，我们证明了有限的最小障碍物列出了3色 $\{P_{Ť}，C_3\}$免费图，当且仅当 $Ť\le 7$。这意味着存在用于列表3着色的多项式时间证明算法$\{P_7，C_3\}$-无图。我们进一步确定其他情况$Ť，\ell$和ķ，从而阻碍最小的家庭名单ķ -coloring在$\{P_{Ť}，C_{\ell }\}$无图是有限的。