On 3-colorability of (claw, diamond)-free graphs

The $3$-colorability problem is a well-known NP-complete problem and it remains NP-complete for $(claw, diamond, K_4)$-free graphs. Recently, $3$-colorability has been also considered for $(claw, N_{1,1,1})$-free graphs. Here, a generalised net $N_{i, j, k}$ is the graph obtained by identifying each vertex of a triangle with an endvertex of one of three vetex-disjont paths of lengths $i, j, k$. We study the class of $(claw, diamond, N_{i, j, k})$-free graphs for $(i, j, k) \in \{(1, 1, 3),$ $(1, 2, 2),$ $(2, 2, 2) \}$. We show that these graphs are $3$-colorable or contain a $K_4$ or belong to some well-defined class of non $3$-colorable graphs. Moreover, we prove that there are only finitely many non $3$-colorable $N_{1, 2, k}$-free graphs for any $k \geq 2$, but there exist infinitely many non $3$-colorable $N_{i, j, k}$-free graphs for any $2 \leq i \leq j \leq k.$