Linkedness of Cartesian products of complete graphs

TL;DR

This paper proves the maximum linkedness of Cartesian products of complete graphs, showing they are loor{(d_1+d_2)/2}-linked, which supports a conjecture about simple polytopes.

Contribution

It establishes the exact linkedness of Cartesian products of complete graphs and connects this to properties of simple polytopes, confirming a specific case of a broader conjecture.

Findings

Cartesian product $K^{d_1+1} imes K^{d_2+1}$ is loor{(d_1+d_2)/2}-linked for $d_1,d_2 ge 2$

The result is optimal and supports the conjecture that all simple $d$-polytopes are loor{d/2}-linked

Provides insight into the structure of graphs of simple polytopes and their linkedness properties.

Abstract

This paper is concerned with the linkedness of Cartesian products of complete graphs. A graph with at least $2k$ vertices is {\it $k$-linked} if, for every set of $2k$ 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 $K^{d_{1}+1}\times K^{d_{2}+1}$ of complete graphs $K^{d_{1}+1}$ and $K^{d_{2}+1}$ is $\floor{(d_{1}+d_{2})/2}$-linked for $d_{1},d_{2}\ge 2$, and this is best possible. %A polytope is said to be {\it $k$-linked} if its graph is $k$-linked. This result is connected to graphs of simple polytopes. The Cartesian product $K^{d_{1}+1}\times K^{d_{2}+1}$ is the graph of the Cartesian product $T(d_{1})\times T(d_{2})$ of a $d_{1}$-dimensional simplex $T(d_{1})$ and a $d_{2}$-dimensional simplex $T(d_{2})$. And the polytope $T(d_{1})\times T(d_{2})$ is a {\it 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 $\floor{d/2}$-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.