Decomposing planar graphs without triangular short cycles into a matching and a 3-colorable graph

Let \(c_1,\ldots , c_k\) $c_1,\dots ,c_k$ be k non-negative integers. A graph G is \((c_1, \ldots , c_k)\) $(c_1,\dots ,c_k)$ -colorable if the vertex set of G can be partitioned into k sets \(V_1, \ldots , V_k\) $V_1,\dots ,V_k$ , such that the induced subgraph \(G[V_i]\) $G[V_i]$ has maximum degree at most \(c_i\) $c_i$ for \(i\in [k]\) $i\in [k]$ . Denote by \(\mathscr {F}\) $\mathcal{F}$ the family of planar graphs without triangles adjacent to cycles of length 5 and 7. This paper proves that if \(G\in \mathscr {F}\) $G\in \mathcal{F}$ , then G is (1, 1, 1)-colorable. As a corollary, every graph in \(\mathscr {F}\) $\mathcal{F}$ can be decomposed into a matching and a 3-colorable graph.