A note on the independence number of 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...

A Note on Triangle-Free Graphs

Triangle-free planar graphs with small independence number

Since planar triangle-free graphs are 3-colourable, such a graph with n vertices has an independent set of size at least n/3. We prove that unless the graph contains a certain obstruction, its independence number is at least n/(3−ε) for some fixed ε > 0. We also provide a reduction rule for this obstruction, which enables us to transform any plane triangle-free graph G into a plane triangle-fre...

Edge number critical triangle free graphs with low independence numbers

The structure of all triangle free graphs G = (V,E) with |E|−6|V |+α(G) = 0 is determined, yielding an affirmative answer to a question of Stanis law Radziszowsky and Donald Kreher.