Unbalanced spanning subgraphs in edge labeled complete graphs

TL;DR

This paper proves the existence of specific subgraphs in edge-labeled complete graphs that have a significantly unbalanced number of edges with a particular label, extending understanding of edge distributions in such graphs.

Contribution

It establishes the existence of isomorphic subgraphs with a biased edge label count, providing bounds that show deviation from randomness in edge label distributions.

Findings

Existence of subgraphs with biased label counts in complete graphs.

Quantitative bounds on the deviation from expected label distribution.

Special results for the case d=1/2 and maximum degree ≤ 2.

Abstract

Let $K$ be a complete graph of order $n$. For $d\in (0,1)$, let $c$ be a $\pm 1$-edge labeling of $K$ such that there are $d{n\choose 2}$ edges with label $+1$, and let $G$ be a spanning subgraph of $K$ of maximum degree at most $\Delta$. We prove the existence of an isomorphic copy $G'$ of $G$ in $K$ such that the number of edges with label $+1$ in $G'$ is at least $\left(c_{d,\Delta}-O\left(\frac{1}{n}\right)\right)m(G)$, where $c_{d,\Delta}=d+\Omega\left(\frac{1}{\Delta}\right)$ for fixed $d$, that is, this number visibly deviates from its expected value when considering a uniformly random copy of $G$ in $K$. For $d=\frac{1}{2}$, and $\Delta\leq 2$, we present more detailed results.