In a recent work, I. Gutman and B. Furtula posed the structure of trees with a single high-degree vertex and smallest ABC index . Here we provide a family of trees with smaller ABC index in one case of their conjectures. The smallest tree violating the Gutman-Furtula conjecture has 312 vertices.
The Erd˝ os-Gyárfás conjecture states that every graph with minimum degree at least three has a cycle whose length is a power of 2. Since this conjecture has proven to be far from reach, Hobbs asked if the Erd˝ os-Gyárfás conjecture holds in claw-free graphs. In this paper, we obtain some results on this question, in particular for cubic claw-free graphs.