On the number of alternating paths in bipartite complete graphs

TL;DR

This paper explores the maximum number of internally disjoint alternating paths of a given length in bipartite complete graphs with edge colorings, introducing the concept of alternating connectivity and employing probabilistic and integer programming methods.

Contribution

It introduces the concept of alternating connectivity, generalizes previous problems on alternating paths, and develops new bounds using random colorings and integer programming techniques.

Findings

Established bounds for maximum t with internally disjoint paths

Analyzed the case without the disjointness constraint

Proposed new methods combining probabilistic and optimization approaches

Abstract

Let $C \subseteq [r]^m$ be a code such that any two words of $C$ have Hamming distance at least $t$. It is not difficult to see that determining a code $C$ with the maximum number of words is equivalent to finding the largest $n$ such that there is an $r$-edge-coloring of $K_{m, n}$ with the property that any pair of vertices in the class of size $n$ has at least $t$ alternating paths (with adjacent edges having different colors) of length $2$. In this paper we consider a more general problem from a slightly different direction. We are interested in finding maximum $t$ such that there is an $r$-edge-coloring of $K_{m,n}$ such that any pair of vertices in class of size $n$ is connected by $t$ internally disjoint and alternating paths of length $2k$. We also study a related problem in which we drop the assumption that paths are internally disjoint. Finally, we introduce a new concept, which we call alternating connectivity. Our proofs make use of random colorings combined with some integer programs.