Discrepancy and Signed Domination in Graphs and Hypergraphs

TL;DR

This paper explores bounds on the signed domination number in graphs and hypergraphs, linking it to discrepancy theory, and provides improved upper and lower bounds over existing results.

Contribution

It introduces new upper and lower bounds for the signed domination number, advancing understanding in graph and hypergraph domination concepts.

Findings

Improved bounds for signed domination number

Enhanced understanding of discrepancy in graphs

Connections between domination and discrepancy theory

Abstract

For a graph G, a signed domination function of G is a two-colouring of the vertices of G with colours +1 and -1 such that the closed neighbourhood of every vertex contains more +1's than -1's. This concept is closely related to combinatorial discrepancy theory as shown by Fueredi and Mubayi [J. Combin. Theory, Ser. B 76 (1999) 223-239]. The signed domination number of G is the minimum of the sum of colours for all vertices, taken over all signed domination functions of G. In this paper, we present new upper and lower bounds for the signed domination number. These new bounds improve a number of known results.