Linkedness of Cartesian products of complete graphs

Abstract

This paper is concerned with the linkedness of Cartesian products of complete graphs. A graph with at least 2 k vertices is k -linked if, for every set of 2 k distinct vertices organised in arbitrary k pairs of vertices, there are k vertex-disjoint paths joining the vertices in the pairs. We show that the Cartesian product K d 1 +1 × K d 2 +1 of complete graphs K d 1 +1 and K d 2 +1 is ⌊( d 1 + d 2 )/2⌋ -linked for d 1 , d 2 ≥ 2 , and this is best possible. This result is connected to graphs of simple polytopes. The Cartesian product K d 1 +1 × K d 2 +1 is the graph of the Cartesian product T ( d 1 ) × T ( d 2 ) of a d 1 -dimensional simplex T ( d 1 ) and a d 2 -dimensional simplex T ( d 2 ) . And the polytope T ( d 1 ) × T ( d 2 ) is a simple polytope , a ( d 1 + d 2 ) -dimensional polytope in which every vertex is incident to exactly d 1 + d 2 edges. While not every d -polytope is ⌊ d /2⌋ -linked, it may be conjectured that every simple d -polytope is. Our result implies the veracity of the revised conjecture for Cartesian products of two simplices.