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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.

The distinguishing chromatic number χ D (G) of a graph G is the least integer k such that there is a proper k-coloring of G which is not preserved by any nontrivial automorphism of G. We study the distinguishing chromatic number of Cartesian products of graphs by focusing on how much it can exceed the trivial lower bound of the chromatic number χ(•). Our main result is that for every graph G, there exists a constant d G such that for all d ≥ d G the distinguishing chromatic number of G d is at most χ(G)+ 1, where G d is the Cartesian product of d copies of G. We also prove that for d ≥ 5, the Cartesian product of d complete graphs has distinguishing chromatic number at most one more than the corresponding chromatic number, and we determine the distinguishing chromatic number of hypercubes exactly.

Two graphs G1 and G2 of order n pack if there exist injective mappings of their vertex sets into [n], such that the images of the edge sets are disjoint. In 1978, Bollobás and Eldridge, and independently Catlin, conjectured that if (∆(G1) + 1)(∆(G2) + 1) ≤ n + 1, then G1 and G2 pack. Towards this conjecture, we show that for ∆(G1), ∆(G2) ≥ 300, if (∆(G1)+1)(∆(G2)+1) ≤ 0.6n+1, then G1 and G2 pack. This is also an improvement, for large maximum degrees, over the classical result by Sauer and Spencer that G1 and G2 pack if ∆(G1)∆(G2) < 0.5n.

In 2003, Kostochka, Pelsmajer, and West introduced a list analogue of equitable coloring called equitable choosability. In this paper, we motivate and define a new list analogue of equitable coloring called proportional choosability. A k-assignment L for a graph G specifies a list L(v) of k available colors for each vertex v of G. An L-coloring assigns a color to each vertex v from its list L(v). For each color c, let η(c) be the number of vertices v whose listWe show that if a graph G is proportionally k-choosable, then every subgraph of G is also proportionally k-choosable and also G is proportionally (k + 1)-choosable, unlike equitable choosability for which analogous claims would be false. We also show that any graph G is proportionally k-choosable whenever k ≥ ∆(G) + ⌈|V (G)|/2⌉, and we use matching theory to completely characterize the proportional choosability of stars and the disjoint union of cliques.