Erdös-Gyárfás Conjecture for Cubic Planar Graphs

Abstract

In 1995, Paul Erdös and András Gyárfás conjectured that for every graph of minimum degree at least 3, there exists a non-negative integer $m$ such that $G$ contains a simple cycle of length $2^m$. In this paper, we prove that the conjecture holds for 3-connected cubic planar graphs. The proof is long, computer-based in parts, and employs the Discharging Method in a novel way.

On 2-power unicyclic cubic graphs

“…A graph G is cubic if every vertex of G is of degree three. More on graphs and cycles can be seen in [3,7,9].By computer searches, Markstrom [6] verified Erdős-Gyárfás conjecture for cubic graphs of order at most 29, and found that the smallest cubic planar graph with no 4or 8-cycles has 24 vertices (see Figure 1). Note that this graph contains a 16-cycle.…”

“…Markstrom's searches found four graphs on 24 vertices in which the only power-of-two cycles have 16 vertices, one of these four graphs is planar. However, the Erdős-Gyárfás conjecture is now known to be true for the special case of 3-connected cubic planar graphs, see Heckman and Krakovski [6]. Weaker results relating the degree of a graph to unavoidable sets of cycle lengths are known.…”

“…Shauger [8] proved the conjecture for K 1,m free graphs of minimum degree at least m + 1 or maximum degree at least 2m − 1. Every 3-connected cubic planar graph contains a 2 m − cycle, for some 2 ≤ m ≤ 7, see [6]. Markstrom's 24-vertex cubic planar graph with no 4 or 8 cycles, found in a computer search for counter example to the Erdős-Gyárfás conjecture, has cycles with 16 vertices.…”