Some Remarks on the Cycle Plus Triangles Problem

Abstract

All (undirected) graphs and digraphs considered are assumed to be finite (if not otherwise stated) and loopless. Multiple edges (arcs) are permitted. For a graph G, let V (G), E(G), and χ(G) denote the vertex set, the edge set, and the chromatic number of G, respectively. If X ⊆ V (G) and F ⊆ E(G), then $G - X - F$ denotes the subgraph H of G satisfying $V (H) = V (G) - X$ and $E(H) =\{ xy\mid xy \in E(G) - F$ and $x,y\not\in X\}$.