The Chvátal–Erdős condition for panconnectivity of triangle-free graphs

Measuring triangle-free graphs

A neighborhood and degree condition for panconnectivity

Let G be a 2-connected graph of order n with x, y ∈ V (G). For u, v ∈ V (G), let Pi[u, v] denote the path with i vertices which connects u and v. In this paper, we prove that if n ≥ 5 and |NG(u)∪NG(v)|+dG(w) ≥ n+1 for every triple of independent vertices u, v, w of G, then there exists a Pi[x, y] in G for 5 ≤ i ≤ n, or G belongs to one of three exceptional classes. This implies a positive answe...

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.

The local density of triangle-free graphs

How dense can every induced subgraph of b nc vertices (0 < 1) of a triangle-free graph of order n be? Tools will be developed to estimate the local density of graphs, based on the spectrum of the graph and on a fractional viewpoint. These tools are used to refute a conjecture of Erd} os et.al. about the local density of triangle-free graphs for a certain range of , by estimating the local densi...

On the Evolution of Triangle-Free Graphs