Almost resolvable even cycle systems of $(K_u \times K_g)(\lambda)$

TL;DR

This paper proves the existence of almost resolvable k-cycle systems in tensor product graphs for all k divisible by 4, with few exceptions, advancing the understanding of cycle decompositions in complex graph structures.

Contribution

It establishes the existence of almost resolvable k-cycle systems in tensor product graphs for all multiples of 4, filling gaps in cycle system theory.

Findings

Almost resolvable k-cycle systems exist for all k ≡ 0 mod 4 in tensor product graphs.

Existence holds for all such systems with few possible exceptions.

The results extend cycle system theory to complex graph products.

Abstract

In this paper, we prove that almost resolvable $k$-cycle systems (briefly $k$-ARCS) of $(K_u \times K_g)(\lambda)$ exists for all $k \equiv 0(mod \ 4) $ with few possible exceptions, where $\times$ represents tensor product of graphs.