A graph is k-degenerate if its vertices can be successively deleted so that when deleted, each has degree at most k. These graphs were introduced by Lick and White in 1970 and have been studied in several subsequent papers. We present sharp bounds on the diameter of maximal k-degenerate graphs and characterize the extremal graphs for the upper bound. We present a simple characterization of the degree sequences of these graphs and consider related results. Considering edge coloring, we conjecture that a maximal k-degenerate graph is class two if and only if it is overfull, and prove this in some special cases. We present some results on decompositions and arboricity of maximal k-degenerate graphs and provide two characterizations of the subclass of k-trees as maximal k-degenerate graphs. Finally, we define and prove a formula for the Ramsey core numbers.

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Plesnik proved that the edge connectivity and minimum degree are equal for diameter 2 graphs. We provide a streamlined proof of this fact and characterize the diameter 2 graphs with a nontrivial minimum edge cut.

The vertex-arboricity a (G) of a graph G is the minimum number of subsets that the vertices of G can be partitioned so that the subgraph induced by each set of vertices is a forest. Kronk and Mitchem proved a generalization of Brooks' Theorem for vertex arboricity, a (G) = 1 + 1 2 △ (G) if and only if G is a cycle or a complete graph of odd order. We provide a short proof of this result using degeneracy.

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