On the Hull Number of Triangle-Free Graphs

On the Hull Number of Triangle-Free Graphs

A set of vertices C in a graph is convex if it contains all vertices which lie on shortest paths between vertices in C. The convex hull of a set of vertices S is the smallest convex set containing S. The hull number h(G) of a graph G is the smallest cardinality of a set of vertices whose convex hull is the vertex set of G. For a connected triangle-free graph G of order n and diameter d ≥ 3, we ...

On the number of pentagons in triangle-free graphs

Using the formalism of flag algebras, we prove that the maximal number of copies of C5 in a triangle-free graph with 5l + a vertices (0 ≤ a ≤ 4) is l(l + 1)a, and we show that the set of extremal graphs for this problem consists precisely of almost balanced blow-ups of a single pentagon. This settles a conjecture made by Erdős in 1984. For the transition from an asymptotic version of our result...

The fractional chromatic number of triangle-free subcubic graphs

Heckman and Thomas conjectured that the fractional chromatic number of any triangle-free subcubic graph is at most 14/5. Improving on estimates of Hatami and Zhu and of Lu and Peng, we prove that the fractional chromatic number of any triangle-free subcubic graph is at most 32/11 ≈ 2.909.

The independent domination number of maximal triangle-free graphs

A triangle-free graph is maximal if the addition of any edge produces a triangle. A set S of vertices in a graph G is called an independent dominating set if S is both an independent and a dominating set of G. The independent domination number i(G) of G is the minimum cardinality of an independent dominating set of G. In this paper, we show that i(G) ≤ δ(G) ≤ n 2 for maximal triangle-free graph...

On the Evolution of Triangle-Free Graphs