Minmax degree of graphs (Extended abstract)

Abstract

Abstract The minmax degree of a graph G, denoted M ⁎ ( G ) , is the largest integer m such that every edge of G is incident with a vertex with degree at least m. The tight bounds are known for the minmax degree of planar graphs with minimum degree at least 2 and girth g for every g, unless when g = 3 or g = 4 . For these last values of g, we can find for any integer k a graph with M ⁎ ( G ) ⩾ k . We define the 2-stackability of a graph G, denoted m a x S 2 ( G ) , as the maximum number of vertices with degree 2 that all are adjacent to the same pair of distinct vertices in G. In the complete version of the paper, we prove that, for a planar graph G with minimum degree at least 2, M ⁎ ( G ) ⩽ 5 ( m a x S 2 ( G ) + 2 ) ; moreover, M ⁎ ( G ) ⩽ 5 ( m a x S 2 ( G ) + 1 ) if G does not contain triangles. We also show that these bounds are tight. The maximum average degree of a graph G is mad ( G ) = max { 2 | E ( G ) | | V ( G ) | ; H ⊆ G } . We prove that, for every δ ⩾ 2 and k ⩾ δ , every graph G with δ ( G ) ⩾ δ and mad ( G ) 2 δ ( k + 1 ) δ + k + 1 has M ⁎ ( G ) ⩽ k . We also show that these bound is the lowest possible. Borodin et al. [Borodin, O. V., A. V. Kostochka, N. N. Sheikh and G. Yu, M-degrees of quadrangle-free planar graph, J. Graph Theory 60 (2009), pp. 80–85] showed that M ⁎ ( G ) ⩽ 7 if G is a planar graph with minimum degree at least 2 and without 4-cycles. Moreover, if G has neither 5-cycles, Borodin et al. [5] proved that M ⁎ ( G ) ⩽ 5 . We prove that a planar graph G has M ⁎ ( G ) ⩽ 5 if G has minimum degree at least 2, no 4-cycles, neither 5-cycles adjacent to a triangle incident to a vertex of degree 2. We give some other results about planar graphs with minimum degree at least 2 and without 4-cycles and intersecting triangles, with other restrictions on the cycles. These results imply results for edge-partitions of graphs and game coloring number.