An equitable coloring of a graph G is a proper vertex coloring of G such that the sizes of any two color classes differ by at most one. In the paper, we pose a conjecture that offers a gap-one bound for the smallest number of colors needed to equitably color every block graph. In other words, the difference between the upper and the lower bounds of our conjecture is at most one. Thus, in some sense, the situation is similar to that of chromatic index, where we have the classical theorem of Vizing and the Andersen–Goldberg–Seymour conjecture for multigraphs. The results obtained in the paper support our conjecture. More precisely, we verify it in the class of block graphs in which each vertex belongs to a maximum independent set. We also show that the conjecture is true for block graphs which contain a vertex that does not lie in an independent set of size larger than two. Finally, we verify the conjecture for some symmetric-like block graphs. In order to derive our results we obtain structural characterizations of block graphs from these classes.

In this paper, we study two closely related problems on bipartite graphs, viz., the partial vertex cover problem and the budgeted maximum coverage problem. Both these problems arise in a number of different application domains, including, but not limited to, computer security and transportation logistics. It is well known that the vertex cover problem is solvable in polynomial time on bipartite graphs. However, the computational complexity of the partial vertex cover problem on bipartite graphs was open, thus far. In this paper, we establish that the partial vertex cover problem is \bf NP-hard, even on bipartite graphs. Our result also establishes that the closely related budgeted maximum coverage problem is \bf NP-hard on bipartite graphs. For the latter problem, we present an $\frac{8}{9}$-approximation algorithm. Our approximation guarantee matches and resolves the integrality gap of the natural linear programming relaxation for this problem and improves upon a recent $\frac{4}{5}$-approximation algorithm for the same problem.