Graphic sequences with a realization containing cycles C3,…,Cℓ

摘要

A non-increasing sequence π=(d1,…,dn) of nonnegative integers is said to be graphic if it is realizable by a simple graph G on n vertices. A graphic sequence π=(d1,…,dn) is said to be potentially 3Cℓ-graphic if there is a realization of π containing cycles of every length r, 3 ≤ r ≤ ℓ. Li et al. proposed a problem about giving a criteria of potentially 3Cℓ-graphic sequences. For ℓ=5,6, Chen et al. investigated this problem and showed that if dℓ≥ℓ2, then π is potentially 3Cℓ-graphic. In this paper, we extend the above results of Chen et al. for ℓ=5,6 to the general case ℓ ≥ 5, and prove that for every integer ℓ ≥ 5, if dℓ≥ℓ2, then π is potentially 3Cℓ-graphic.