3-minimal triangle-free graphs

On minimal triangle-free 6-chromatic graphs

A graph with chromatic number k is called k-chromatic. Using computational methods, we show that the smallest triangle-free 6-chromatic graphs have at least 32 and at most 40 vertices. We also determine the complete set of all triangle-free 5-chromatic graphs up to 23 vertices and all triangle-free 5-chromatic graphs on 24 vertices with maximum degree at most 7. This implies that Reed’s conject...

On-Line 3-Chromatic Graphs I. Triangle-Free Graphs

This is the first half of a two-part paper devoted to on-line 3-colorable graphs. Here on-line 3-colorable triangle-free graphs are characterized by a finite list of forbidden induced subgraphs. The key role in our approach is played by the family of graphs which are both triangleand (2K2 + K1)-free. Characterization of this family is given by introducing a bipartite modular decomposition conce...

Triangle-Free Outerplanar 3-Graphs Are Pairwise Compatibility Graphs

A graph G = (V,E) is called a pairwise compatibility graph (PCG) if there exists an edge-weighted tree T and two non-negative real numbers dmin and dmax such that each vertex u ′ ∈ V corresponds to a leaf u of T and there is an edge (u′, v′) ∈ E if and only if dmin ≤ dT (u, v) ≤ dmax in T . Here, dT (u, v) denotes the distance between u and v in T , which is the sum of the weights of the edges ...

Measuring triangle-free graphs

Triangle-free Uniquely 3-Edge Colorable Cubic Graphs

This paper presents infinitely many new examples of triangle-free uniquely 3-edge colorable cubic graphs. The only such graph previously known was given by Tutte in 1976.