Zero-Sum Copies of Spanning Forests in Zero-Sum Complete Graphs

Abstract

For a complete graph \(K_n\) of order n, an edge-labeling \(c:E(K_n)\rightarrow \{ -1,1\}\) 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 \(|c(E(F'))|\le \Delta (F)+1\), where \(\Delta (F)\) is the maximum degree of F. We conjecture that this bound can be improved to \(|c(E(F'))|\le (\Delta (F)-1)/2\) 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.