On existentially complete triangle-free graphs

Triply Existentially Complete Triangle-Free Graphs

A triangle-free graph G is called k-existentially complete if for every induced k-vertex subgraph H of G, every extension of H to a (k+ 1)-vertex triangle-free graph can be realized by adding another vertex of G to H. Cherlin [8, 9] asked whether k-existentially complete triangle-free graphs exist for every k. Here we present known and new constructions of 3existentially complete triangle-free ...

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...