A stability theorem for maximal K r+1 -free graphs

A Generalization of Dirac's Theorem for K(1,3)-free Graphs

It is known that ff a 2-connected graph G of suiBciently large order n satisfies the property that the union of the neighborhoods of each pair of vertices has cardinallty at least ~-, " then G is hamiltonian. In this paper, we obtain a similar generalization of Dirac's Theorem for K(1, 3)-free graphs. In particular, we show that if G is a 2-connected K(1, 3)-free graph of order n with the cardi...

A Brooks' Theorem for Triangle-Free Graphs

Let G be a triangle-free graph with maximum degree ∆(G). We show that the chromatic number χ(G) is less than 67(1 + o(1))∆/ log∆.

Cycle-maximal triangle-free graphs

We conjecture that the balanced complete bipartite graph K⌊n/2⌋,⌈n/2⌉ contains more cycles than any other n-vertex triangle-free graph, and we make some progress toward proving this. We give equivalent conditions for cycle-maximal triangle-free graphs; show bounds on the numbers of cycles in graphs depending on numbers of vertices and edges, girth, and homomorphisms to small fixed graphs; and u...

Random maximal H-free graphs

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...