Potentially K m — G-graphical sequences: A survey

Abstract

The set of all non-increasing nonnegative integer sequences π = (d(v 1), d(v 2), …, d(v n )) is denoted by NS n . A sequence π ∈ NS n is said to be graphic if it is the degree sequence of a simple graph G on n vertices, and such a graph G is called a realization of π. The set of all graphic sequences in NS n is denoted by GS n . A graphical sequence π is potentially H-graphical if there is a realization of π containing H as a subgraph, while π is forcibly H-graphical if every realization of π contains H as a subgraph. Let K k denote a complete graph on k vertices. Let K m −H be the graph obtained from Km by removing the edges set E(H) of the graph H (H is a subgraph of K m ). This paper summarizes briefly some recent results on potentially K m −G-graphic sequences and give a useful classification for determining σ (H, n).