Degree sequences in triangle-free graphs

Degree Sequences of F-Free Graphs

Let F be a fixed graph of chromatic number r + 1. We prove that for all large n the degree sequence of any F -free graph of order n is, in a sense, close to being dominated by the degree sequence of some r-partite graph. We present two different proofs: one goes via the Regularity Lemma and the other uses a more direct counting argument. Although the latter proof is longer, it gives better esti...

Measuring triangle-free graphs

Median Graphs and Triangle-Free Graphs

Let M(m,n) be the complexity of checking whether a graph G with m edges and n vertices is a median graph. We show that the complexity of checking whether G is triangle-free is at most M(m,m). Conversely, we prove that the complexity of checking whether a given graph is a median graph is at most O(m log n) + T (m log n, n), where T (m,n) is the complexity of finding all triangles of the graph. W...

Line graphs of triangle-free graphs

We present some results on the structure of line graphs of triangle-free graphs and discuss connections with homogeneity.

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...