Vector Choosability in Bipartite Graphs

Abstract

The vector choice number of a graph G over a field \(\mathbb {F}\), introduced by Haynes et al. (Electron. J. Comb., 2010), is the smallest integer k such that for every assignment of k-dimensional subspaces of a vector space over \(\mathbb {F}\) to the vertices, it is possible to choose nonzero vectors for the vertices from their subspaces so that the vectors received by adjacent vertices are orthogonal over \(\mathbb {F}\). This work is concerned with the vector choice number of bipartite graphs over various fields. We first observe that the vector choice number of bipartite graphs can be arbitrarily large over any field. We then consider the problem of estimating, for a given integer k, the smallest integer m for which the vector choice number of the complete bipartite graph \(K_{k,m}\) over \(\mathbb {F}\) exceeds k. We prove upper and lower bounds on this quantity, implying a substantial difference between the behavior of the (color) choice number and the vector choice number on bipartite graphs. For the computational aspect, we show a hardness result for deciding whether the vector choice number of a given bipartite graph over \(\mathbb {F}\) is at most k, provided that \(k \ge 3\) and that \(\mathbb {F}\) is either the real field or any finite field.