Oriented discrepancy of Hamilton cycles and paths in digraphs

TL;DR

This work extends oriented discrepancy theory to directed graphs by proposing two Ore-type conjectures about Hamilton oriented cycles with many forward arcs. It develops degree-sum and neighborhood-structure results that support these conjectures, along with approximate bounds based on $s^*(D)$, and connects these objectives to costs of path-cycle and cycle factors in a symmetric digraph $\hat{D}$. For semicomplete multipartite digraphs, it provides polynomial-time algorithms to maximize forward-arcs on Hamilton oriented paths and cycles via cost-optimization in $\hat{D}$, and extends similar tractable characterizations to locally semicomplete digraphs with explicit component-structure-based procedures. The paper also delineates algorithmic limits by proving NP-hardness for maximum-forward-arc Hamilton cycles in several natural digraph classes, clarifying the boundaries of efficient computation in directed discrepancy problems.

Abstract

Erd{\H o}s (1963) initiated extensive graph discrepancy research on 2-edge-colored graphs. Gishboliner, Krivelevich, and Michaeli (2023) launched similar research on oriented graphs. They 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. This conjecture was proved by Freschi and Lo (2024) who posed an open problem to extend their result to an Ore-type condition. We propose two conjectures for such extensions and prove some results which provide support to the conjectures. For forward arc maximization on Hamilton oriented cycles and paths in semicomplete multipartite digraphs and locally semicomplete digraphs, we obtain characterizations which lead to polynomial-time algorithms.

Figures (4)

• Figure 1: Tight examples for Conjecture \ref{['conj:1']}
• Figure 2: Tight examples for Conjecture \ref{['conj:2']}
• Figure 3: The four good diamonds
• Figure 4: 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 (43)

• Theorem 1.1
• Theorem 1.2: FL24
• Conjecture 1.3
• Conjecture 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 2.1