An n-tuple π (not necessarily monotone) is graphic if there is a simple graph G with vertex set {v 1 , . . . , v n } in which the degree of v i is the ith entry of π. Graphic n-tuples (d (1) 1 , . . . , d(1)for all i. We prove that graphic n-tuples π 1 and π 2 pack if ∆ ≤ √ 2δn − (δ − 1), where ∆ and δ denote the largest and smallest entries in π 1 + π 2 (strict inequality when δ = 1); also, the bound is sharp.Kundu and Lovász independently proved that a graphic n-tuple π is realized by a graph with a k-factor if the n-tuple obtained by subtracting k from each entry of π is graphic; for even n we conjecture that in fact some realization has k edge-disjoint 1-factors. We prove the conjecture in the case where the largest entry of π is at most n/2 + 1 and also when k ≤ 3.

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A cycle C of length k is extendable if there is a cycle C of length k + 1 with V (C) ⊂ V (C). A graph G = (V, E) of order n is cycle extendable when every cycle C of length k < n is extendable. A chordal graph is a spider intersection graph if it admits an intersection representation which consists of subtrees of a subdivided star (or spider). In 1990, Hendry conjectured that all hamiltonian chordal graphs are cycle extendible, and this conjecture remains unresolved. We show that all hamiltonian spider intersection graphs are cycle extendable, generalizing known results on cycle extendability in interval graphs and split graphs.

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