Zero-sum copies of spanning forests in zero-sum complete graphs

TL;DR

This paper investigates conditions under which spanning forests in zero-sum edge-labeled complete graphs can have zero or near-zero total edge sum, providing bounds and verifying conjectures for specific forest types.

Contribution

It establishes a bound on the edge sum for spanning forests in zero-sum complete graphs and verifies a conjecture for star forests, also exploring zero-sum factors under divisibility conditions.

Findings

Existence of a spanning forest copy with edge sum at most Δ(F)+1.

Verification of the conjecture for star forests K_{1,n-1}.

Conditions for zero-sum P_3- and P_4-factors in large graphs.

Abstract

For a complete graph $K_n$ of order $n$, an edge-labeling $c:E(K_n)\to \{ -1,1\}$ satisfying $c(E(K_n))=0$, and a spanning forest $F$ of $K_n$, we consider the problem to minimize $|c(E(F'))|$ over all isomorphic copies $F'$ of $F$ in $K_n$. In particular, we ask under which additional conditions there is a zero-sum copy, that is, a copy $F'$ of $F$ with $c(E(F'))=0$. We show that there is always a copy $F'$ of $F$ with $|c(E(F'))|\leq \Delta(F)+1$, where $\Delta(F)$ is the maximum degree of $F$. We conjecture that this bound can be improved to $|c(E(F'))|\leq (\Delta(F)-1)/2$ and verify this for $F$ being the star $K_{1,n-1}$. Under some simple necessary divisibility conditions, we show the existence of a zero-sum $P_3$-factor, and, for sufficiently large $n$, also of a zero-sum $P_4$-factor.