arXiv subjects
Publications and source records attributed to Frederic Maffray.
For a graph $G$, let $\chi(G)$ and $\omega(G)$ respectively denote the chromatic number and clique number of $G$. We give an explicit structural description of ($P_5$,gem)-free graphs, and show that every such graph $G$ satisfies $\chi(G)\le \lceil\frac{5\omega(G)}{4}\rceil$. Moreover, this bound is best possible.
We elucidate the structure of $(P_6,C_4)$-free graphs by showing that every such graph either has a clique cutset, or a universal vertex, or belongs to several special classes of graphs. Using this result, we show that for any $(P_6,C_4)$-free graph $G$, $\lceil\frac{5\omega(G)}{4}\rceil$ and $\lceil\frac{\Delta(G) + \omega(G) +1}{2}\rceil$ are tight upper bounds for the chromatic number of $G$. Moreover, our structural results imply that every ($P_6$,$C_4$)-free graph with no clique cutset has bounded clique-width, and thus the existence of a polynomial-time algorithm that computes the chromatic number (or stability number) of any $(P_6,C_4)$-free graph.
We consider the class of Berge graphs that do not contain a chordless cycle of length $4$. We present a purely graph-theoretical algorithm that produces an optimal coloring in polynomial time for every graph in that class.
A bull is a graph with five vertices $r, y, x, z, s$ and five edges $ry$, $yx$, $yz$, $xz$, $zs$. A graph $G$ is bull-reducible if no vertex of $G$ lies in two bulls. We prove that every bull-reducible Berge graph $G$ that contains no antihole is weakly chordal, or has a homogeneous set, or is transitively orientable. This yields a fast polynomial time algorithm to color exactly the vertices of such a graph.