In online list coloring (introduced by Zhu and by Schauz in 2009), on each round the set of vertices having a particular color in their lists is revealed, and the coloring algorithm chooses an independent subset to receive that color. The paint number of a graph G is the least k such that there is an algorithm to produce a successful coloring with no vertex being shown more than k times; it is at least the choice number. We study paintability of joins with complete or empty graphs, obtaining a partial result toward the paint analogue of Ohba's Conjecture. We also determine upper and lower bounds on the paint number of complete bipartite graphs and characterize 3-paintcritical graphs.

Using the existence of noncrossing Eulerian circuits in Eulerian plane graphs, we give a short constructive proof of the theorem of Heawood that Eulerian triangulations are 3-colorable.

Let A be a sequence of natural numbers, r A (n) be the number of ways to represent n as a sum of consecutive elements in A, and M A (x) := P n≤x r A (n). We give a new short proof of LeVeque's formula regarding M A (x) when A is an arithmetic progression, and then extend the proof to give asymptotic formulas for the case when A behaves almost like an arithmetic progression, and also when A is the set of primes in an arithmetic progression.