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A well-known combinatorial theorem says that a set of n non-collinear points in the plane determines at least n distinct lines. Chen and Chvatal conjectured that this theorem extends to metric spaces, with an appropriated definition of line. In this work we prove a slightly stronger version of Chen and Chvatal conjecture for a family of graphs containing chordal graphs and distance-hereditary graphs.
Tuza famously conjectured in 1981 that in a graph without k+1 edge-disjoint triangles, it suffices to delete at most 2k edges to obtain a triangle-free graph. The conjecture holds for graphs with small treewidth or small maximum average degree, inclu
A $k$-linear coloring of a graph $G$ is an edge coloring of $G$ with $k$ colors so that each color class forms a linear forest -- a forest whose each connected component is a path. The linear arboricity $chi_l(G)$ of $G$ is the minimum integer $k$ su
A propositional logic sentence in conjunctive normal form that has clauses of length two (a 2-CNF) can be associated with a multigraph in which the vertices correspond to the variables and edges to clauses. We first show that every such sentence that
Tuza (1981) conjectured that the size $tau(G)$ of a minimum set of edges that intersects every triangle of a graph $G$ is at most twice the size $ u(G)$ of a maximum set of edge-disjoint triangles of $G$. In this paper we present three results regard
We prove a conjecture of Ohba which says that every graph $G$ on at most $2chi(G)+1$ vertices satisfies $chi_ell(G)=chi(G)$.