Beatriz Martins

Beatriz Martins, Nicolas Trotignon

We revisit a classical paper about (even hole, triangle)-free graphs [Conforti, Cornuéjols, Kapoor and Vu\v sković, Triangle-free graphs that are signable without even holes, Journal of Graph Theory, 34(3), 204--220, 2000]. In fact, the previous study describes a more general class, the so called triangle-free odd signable graphs, and we further generalise the class to the (theta, triangle, wac)-free graphs (not worth defining in an abstract). We exhibit a stronger structure theorem, by precisely describing basic classes and separators. We prove that the separators preserve the treewidth and several properties. Some consequences are a recognition algorithm with running time $O(|V(G)|^4|E(G)|)$, a proof that the treewidth of graphs in the class is at most~4 (improving a previous bound of~5), and a simple criterion to decide if a graph in the class is planar.

Júlio Araújo, Manoel Campêlo, Beatriz Martins et al.

A proper coloring $c$ of a simple graph $G$ is harmonious if, for every pair of distinct edges $uv,xy\in E(G)$, we have that $\{c(u),c(v)\}\neq \{c(x),c(y)\}$. The harmonious chromatic number of $G$, denoted by $h(G)$, is the least positive integer $k$ such that $G$ has a harmonious coloring with $k$ colors. In this work, we extend an idea presented in [Kolay, et al. Harmonious coloring: Parameterized algorithms and upper bounds. Theor. Comp. Sci. 772 (2019), 132-142] to compare the harmonious chromatic numbers of two graphs $G$ and $H$, with $H$ being obtained from $G$ by identifying vertices at distance at least three. Furthermore, by fixing a proof presented in the same work, we manage to improve one of its upper bounds. We also introduce and study the first, to the best of our knowledge, integer-linear programming formulations for this problem in the literature, along with some heuristics. We provide some preliminary tests on random instances and instances from the second DIMACS Implementation Challenge.