Sufficient conditions for planar graphs without 4-cycles and 5-cycles to be 2-degenerate

Abstract

A graph G is k-degenerate if every subgraph of G has a vertex of degree at most k. It is known that every planar graph of girth 6 (equivalently, without 3-, 4-, and 5-cycles) is 2-degenerate. On the other hand, there exist planar graphs without 4 and 5-cycles such as a truncated dodecahedral graph that are not 2-degenerate. Furthermore, a truncated dodecahedral graph also contains none of 6-, 7-, 8-, and 9-cycles. This motivates us to find sufficient conditions for planar graphs without 4-cycles and 5-cycles to be 2-degenerate.In this work, we investigate the degeneracy of planar graphs without 4-, 5-, j-, and k-cycles where j∈{6,7,8,9} and k∈{10,11}. For each j∈{6,9}, we give an example of a 3-regular planar graph without 4-, 5-, j-, 10-, and 11-cycles. In contrast, we prove that every planar graph without 4-, 5-, j-, and k-cycles is 2-degenerate for each j∈{7,8} and k∈{10,11}.