Guessing Games on Triangle-Free Graphs

Guessing Games on Triangle-Free Graphs

The guessing game introduced by Riis [Electron. J. Combin. 2007] is a variant of the “guessing your own hats” game and can be played on any simple directed graph G on n vertices. For each digraph G, it is proved that there exists a unique guessing number gn(G) associated to the guessing game played on G. When we consider the directed edge to be bidirected, in other words, the graph G is undirec...

On Generating Triangle-Free Graphs

We show that the problem to decide whether a graph can be made triangle-free with at most k edge deletions remains NP-complete even when restricted to planar graphs of maximum degree seven. In addition, we provide polynomial-time data reduction rules for this problem and obtain problem kernels consisting of 6k vertices for general graphs and 11k/3 vertices for planar graphs.

Measuring triangle-free graphs

A note on triangle-free graphs

We show that if G is a simple triangle-free graph with n ≥ 3 vertices, without a perfect matching, and having a minimum degree at least n−1 2 , then G is isomorphic either to C5 or to Kn−1 2 , n+1 2 .

A note on maximal triangle-free graphs

We show that a maximal triangle-free graph on n vertices with minimum degree δ contains an independent set of 3δ − n vertices which have identical neighborhoods. This yields a simple proof that if the binding number of a graph is at least 3/2 then it has a triangle. This was conjectured originally by Woodall. We consider finite undirected graphs on n vertices with minimum degree δ. A maximal tr...