A variation of a classical Turán-type extremal problem

  • Authors:

  • Jian-Hua Yin;Jiong-Sheng Li;Guo-Liang Chen

  • Affiliations:

  • Department of Applied Mathematics, College of Information Science and Technology, Hainan University, Haikou, Hainan 570228, China;Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China;Department of Computer Science and Technology, University of Science and Technology of China, Hefei, Anhui 230027, China

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2004

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Abstract

A variation of a classical Turán-type extremal problem (Erdös on Graphs: His Legacy of Unsolved Problems (1998) p. 36) is considered as follows: determine the smallest even integer σ (Kr,s,n) such that every n-term graphic non-increasing sequence π = (d1, d2,...,dn) with term sum σ(π) = d1 + d2 +...+ dn ≥ σ(Kr,s, n) has a realization G containing Kr,s as a subgraph, where Kr,s, is a r × s complete bipartite graph. In this paper, we determine σ(Kr,s, n) exactly for every fixed s ≥ r ≥ 3 when n ≥ n0(r,s), where m = [(r+s+1)2/4] and n0(r,s) = {m + 3s2 - 2s - 6, if s ≤ 2r and s is even, m + 3s2 + 2s - 8, if s ≤ 2r and s is odd, m + 2s2 + (2r - 6)s + 4r - 8, if s ≤ 2r+1.