Modulo flow is a powerful tool in the study of flows in both ordinary graphs and signed graphs. For ordinary graphs, Tutte showed that a graph admits a nowhere-zero -flow if and only if it admits a nowhere-zero -flow. However, such equivalence does not hold any more for signed graphs. Mačajova and Škoviera [SIAM Journal of Discrete Mathematics 31 (2017) 1937–1952] proved that every flow-admissible signed graph with a nowhere-zero -flow admits a nowhere-zero 4-flow. DeVos et al. [Journal of Combinatorial Theory Series B 149 (2021) 198–221] proved that every flow-admissible signed graph admits a nowhere-zero 11-flow by converting certain special nowhere-zero -flows into integer flows, and as a key step, they showed that every signed graph with a nowhere-zero -flow admits a nowhere-zero 5-flow. In this paper, we study how to convert -flows and
In 2023, Bang-Jensen, Havet and Yeo [J. Graph Theory 102 (2023) 578-606] conjectured that every -strong semicomplete digraph contains a hamiltonian cycle avoiding any prescribed set of arcs, which was inspired by the result of Fraisse and Thomassen that, every -strong tournament contains a hamiltonian cycle avoiding any prescribed set of arcs. In this paper, we prove the conjecture.
We exhibit non-switching-isomorphic signed graphs that share a common underlying graph and common chromatic polynomials, thereby answering a question posed by Zaslavsky. We introduce a new pair of bivariate chromatic polynomials that generalises the chromatic polynomials of signed graphs. We establish recursive dominating-vertex deletion formulae for these bivariate chromatic polynomials. As an application, we demonstrate that for a certain family of signed threshold graphs, isomorphism can be characterised by the equality of bivariate chromatic polynomials.
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