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A rainbow matching in an edge-colored graph is a matching in which all the edges have distinct colors. Wang asked if there is a function f (δ) such that a properly edgecolored graph G with minimum degree δ and order at least f (δ) must have a rainbow matching of size δ. We answer this question in the affirmative; f (δ) = 6.5δ suffices. Furthermore, the proof provides a O(δ(G)|V (G)| 2 )-time algorithm that generates such a matching.

A sequence π = (d 1 ,. .. , dn) is graphic if there is a simple graph G with vertex set {v 1 ,. .. , vn} such that the degree of v i is d i. We say that graphic sequences π 1 = (d (1) 1 ,. .. , d (1) n) and π 2 = (d (2) 1 ,. .. , d (2) n) pack if there exist edge-disjoint n-vertex graphs G 1 and G 2 such that for j ∈ {1, 2}, d G j (v i) = d (j) i for all i ∈ {1,. .. , n}. Here, we prove several extremal degree sequence packing theorems that parallel central results and open problems from the graph packing literature. Specifically, the main result of this paper implies degree sequence packing analogues to the Bollobás-Eldridge-Catlin graph packing conjecture and the classical graph packing theorem of Sauer and Spencer. In discrete tomography, a branch of discrete imaging science, the goal is to reconstruct discrete objects using data acquired from low-dimensional projections. Specifically, in the k-color discrete tomography problem the goal is to color the entries of an m × n matrix using k colors so that each row and column receives a prescribed number of entries of each color. This problem is equivalent to packing the degree sequences of k bipartite graphs with parts of sizes m and n. Here we also prove several Sauer-Spencer-type theorems with applications to the two-color discrete tomography problem.

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The strong chromatic index of a graph G, denoted χ ′ s (G), is the least number of colors needed to edge-color G so that edges at distance at most two receive distinct colors. The strong list chromatic index, denoted χ ′ ℓ,s (G), is the least integer k such that if arbitrary lists of size k are assigned to each edge then G can be edge-colored from those lists where edges at distance at most two receive distinct colors. We use the discharging method, the Combinatorial Nullstellensatz, and computation to show that if G is a subcubic planar graph with girth(G) ≥ 41 then χ ′ ℓ,s (G) ≤ 5, answering a question of Borodin and Ivanova [Precise upper bound for the strong edge chromatic number of sparse planar graphs, Discuss. Math. Graph Theory, 33(4), (2014) 759-770]. We further show that if G is a subcubic planar graph and girth(G) ≥ 30, then χ ′ s (G) ≤ 5, improving a bound from the same paper. Finally, if G is a planar graph with maximum degree at most four and girth(G) ≥ 28, then χ ′ s (G) ≤ 7, improving a more general bound of Wang and Zhao from [Odd graphs and its application on the strong edge coloring, arXiv:1412.8358] in this case.