Perfect Matchings in Grid Graphs after Vertex Deletions

Abstract

We consider the d-dimensional grid graph $G=G_m^d$ on vertices $\{1,2,\ldots ,m\}^d$ (a subset of ${\bf Z}^d$), where two vertices are joined if and only if their coordinates differ in one place and have a difference of just 1. The graph is bipartite, and the $m^d$ vertices have bipartition W and B (sets W, B can be determined by the parity of their sum of coordinates). We show that there are constants $a_d,b_d$ so that for every even m, if we choose subsets $B'\subseteq B$ and $W'\subseteq W$ in the d-dimensional grid graph G, which satisfy the three conditions (i) $|B'|=|W'|$, (ii) for any $x,y\in B'$, $d_G(x,y)\ge a_dm^{1/d}+b_d$, and (iii) for any $x,y\in W'$, $d_G(x,y)\ge a_dm^{1/d}+b_d$, then G with the vertices $B'\cup W'$ deleted has a perfect matching. The factor $m^{1/d}$ is best possible.