ON THE EXISTENCE OF CYCLES WITH LENGTH A POWER OF 2 IN CLAW-FREE GRAPHS

The Erd¨os-Gy´arf´as 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 shown itself far from reach, Hobbs asked if the Erd¨os-Gy´arf´as conjecture holds in claw-free graphs. In this paper, we prove that every claw-free graph with the minimum degree at least 4, which does not have C4 as a subgraph, has a cycle whose length is a power of 2 on each of its non-cut vertices.