Unbalanced Spanning Subgraphs in Edge Labeled Complete Graphs

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 $\Delta$ and with $m(G)$ edges. 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 $\left(d+\frac{\min\left\{ 2-d-2\sqrt{1-d},\sqrt{d}-d\right\}}{2\Delta+1}-O\left(\frac{1}{n}\right)\right)m(G)$, that is, this number visibly exceeds its expected value $d\cdot m(G)$ when considering a uniformly random copy of $G$ in $K$. For $d=\frac{1}{2}$, and $\Delta\leq 2$, we present more detailed results.