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On the Order of a Graph with a given Degree Set

Tarandeep Singh Ahuja1, Amitabha Tripathi2
129 Public Park, Sri Ganganagar ~ 335 001, Rajasthan, India
2Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi — 110 016, India

Abstract

The degree set of a finite simple graph G is the set of distinct degrees of vertices of G. For any given finite set D of positive integers, we determine all positive integers n such that D is the degree set of some simple graph with n vertices. This extends a theorem of Kapoor, Polimeni \& Wall (1977) which shows that the least such n is 1+max(D).

Keywords: Degree sequence, degree set, graphic sequence, matching