Triangle decompositions of $\lambda K_v-\lambda K_w-\lambda K_u$

TL;DR

This paper characterizes when certain complete multi-graphs with edges removed can be decomposed into triangles, providing necessary and sufficient conditions for odd multiplicities and conditions for large even multiplicities.

Contribution

It offers a complete characterization of triangle decompositions for graphs formed by removing two disjoint complete subgraphs from a complete multi-graph, extending previous results.

Findings

For odd λ, precise necessary and sufficient conditions are given.

For even λ, large v ensures conditions are sufficient.

Provides a complete solution for triangle decompositions in these graphs.

Abstract

Denote by $\lambda K_v$ the complete graph of order $v$ with multiplicity $\lambda$. Let $\lambda K_v-\lambda K_w-\lambda K_u$ be the graph obtained from $\lambda K_v$ by the removal of the edges of two vertex disjoint complete multi-subgraphs with multiplicity $ \lambda $ of orders $ w $ and $ u $, respectively. When $\lambda$ is odd, it is shown that there exists a triangle decomposition of $\lambda K_v-\lambda K_w-\lambda K_u$ if and only if $v\geq w+u+\max\{u,w\}$, $ \lambda \left({v\choose 2}-{u\choose 2}-{w\choose 2}\right) \equiv 0 \pmod 3$ and $\lambda (v-w) \equiv \lambda (v-u) \equiv \lambda (v-1) \equiv 0 \pmod 2$. When $\lambda$ is even, it is shown that for large enough $v$, the elementary necessary conditions for the existence of a triangle decomposition of $\lambda K_v-\lambda K_w-\lambda K_u$ are also sufficient.