The optimal χ\chi-bound for (P7,C4,C5)(P_7,C_4,C_5)-free graphs

In this paper, we give an optimal $\chi$-binding function for the class of $(P_7,C_4,C_5)$-free graphs. We show that every $(P_7,C_4,C_5)$-free graph $G$ has $\chi(G)\le \lceil \frac{11}{9}\omega(G) \rceil$. To prove the result, we use a decomposition theorem obtained in [K. Cameron and S. Huang and I. Penev and V. Sivaraman, The class of $({P}_7,{C}_4,{C}_5)$-free graphs: Decomposition, algorithms, and $\chi$-boundedness, Journal of Graph Theory 93, 503--552, 2020] combined with careful inductive arguments and a nontrivial use of the K\"{o}nig theorem for bipartite matching.Comment: 21 page

Near Optimal Colourability on Hereditary Graph Families

A graph family $\mathcal{G}$ is near optimal colourable if there is a constant number $c$ such that every graph $G\in\mathcal{G}$ satisfies $\chi(G)\leq\max\{c,\omega(G)\}$, where $\chi(G)$ and $\omega(G)$ are the chromatic number and clique number of $G$, respectively. The near optimal colourable graph families together with the Lov{\'a}sz theta function are useful for the study of the chromatic number problems for hereditary graph families. In this paper, we investigate the near optimal colourability for ($H_1,H_2$)-free graphs. Our main result is an almost complete characterization for the near optimal colourability for ($H_1,H_2$)-free graphs with two exceptional cases, one of which is the celebrated Gy{\'a}rf{\'a}s conjecture. To obtain the result, we prove that the family of ($2K_2,P_4\vee K_n$)-free graphs is near optimal colourable for every positive integer $n$ by inductive arguments.Comment: 11 pages, 1 figur

Complexity of C_k-Coloring in Hereditary Classes of Graphs

For a graph F, a graph G is F-free if it does not contain an induced subgraph isomorphic to F. For two graphs G and H, an H-coloring of G is a mapping f:V(G) -> V(H) such that for every edge uv in E(G) it holds that f(u)f(v)in E(H). We are interested in the complexity of the problem H-Coloring, which asks for the existence of an H-coloring of an input graph G. In particular, we consider H-Coloring of F-free graphs, where F is a fixed graph and H is an odd cycle of length at least 5. This problem is closely related to the well known open problem of determining the complexity of 3-Coloring of P_t-free graphs. We show that for every odd k >= 5 the C_k-Coloring problem, even in the precoloring-extension variant, can be solved in polynomial time in P_9-free graphs. On the other hand, we prove that the extension version of C_k-Coloring is NP-complete for F-free graphs whenever some component of F is not a subgraph of a subdivided claw