On $t$-edge-balanced graphs

TL;DR

This paper investigates the existence of $t$-edge-balanced graphs, providing new examples for $t=3$ and proving nonexistence for $t \\ge 4$, thus advancing understanding of graph substructure symmetry.

Contribution

It introduces the first known examples of 3-edge-balanced graphs and establishes nonexistence results for t- edge-balanced graphs when t \\ge 4.

Findings

Found the first examples of 3-edge-balanced graphs.

Proved no nontrivial t-edge-balanced graphs exist for t \\ge 4.

Abstract

A graph $G$ on $n$ vertices with $k$ edges is $t$-edge-balanced if every graph on $n$ vertices with $t$ edges is contained in exactly the same number of subgraphs of $K_n$ isomorphic to $G$. Despite the existence of infinite families of $2$-edge-balanced graphs, no $t$-edge-balanced graphs were known for $t \ge 3$. This paper resolves the existence question for $t \ge 3$ in two directions. For $t = 3$, we derive necessary arithmetic conditions on the parameters $(n,k)$ and use a simulated annealing search to find the first known examples of $3$-edge-balanced graphs. For $t \ge 4$, we prove that no nontrivial $t$-edge-balanced graphs exist.