Forward Arc Maximization for Hamilton Oriented Cycles and Paths in Generalizations of Tournaments

TL;DR

This work studies the problem MFAHOC/MFAHOP—maximizing forward arcs in Hamilton oriented cycles and paths—in generalizations of tournaments, focusing on semicomplete multipartite and locally semicomplete digraphs. By reducing the objective to maximum-cost 1-path-cycle (and cycle) factors in the symmetric digraph $\hat{D}$, the authors derive exact characterizations for the maximum forward-arc totals $\sigma^{hp}_{\max}$ and $\sigma^{hc}_{\max}$, and provide polynomial-time algorithms to obtain optimal Hamiltonian structures. For semicomplete multipartite digraphs under HP-majority/HC-majority inequalities, they prove $\sigma^{hp}_{\max}=c^{pc}_{\max}$ and $\sigma^{hc}_{\max}=c^{cf}_{\max}$ (with a single exception yielding $\sigma^{hc}_{\max}=n-1$), while in locally semicomplete digraphs they give a complete structural characterization of the maximum forward-arc Hamilton cycles and an efficient construction. The results advance directed discrepancy theory by delivering concrete, computable criteria for optimum forward arcs in structured digraphs, and they also delineate the boundary between tractability and NP-hardness in broader generalizations where the Hamilton cycle problem remains polynomial-time solvable in some subclasses but the MFAHOC/MFAHOP problem does not.

Abstract

Gishboliner, Krivelevich, and Michaeli (2023) conjectured the following generalization of Dirac's theorem: If the minimum degree $δ$ of an $n$-vertex oriented graph $G$ is greater or equal to $n/2$, then $G$ has a Hamilton oriented cycle with at least $δ$ forward arcs. Freschi and Lo (2024) proved this conjecture. In this paper, we study the problem of maximizing the number of forward arcs in Hamilton oriented cycles/paths in generalizations of tournaments. We obtain characterizations for the maximum number of forward arcs in semicomplete multipartite digraphs and locally semicomplete digraphs. These characterizations lead to polynomial-time algorithms. Note that the above problems are NP-hard for some other generalizations of tournaments even though the Hamilton cycle problem is polynomial-time solvable for these digraph classes.

Figures (1)

• Figure 1: Arcs from cycle $C_i$ to $C_j$, for $1 \leq i < j \leq 3$ are not shown. Note that $C_1 \rightsquigarrow_{V_1} C_2$ and $C_2 \rightsquigarrow_{V_3} C_3$. As there are no arcs from $C_3$ to $C_1$, we have $C_1 \rightsquigarrow_{V_i} C_3$ for all $i=1,2,3$.

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Proposition 2.1
• proof
• proof
• Theorem 2.2: BGY98Y99