Proper connection and proper-walk connection of digraphs

TL;DR

This paper investigates the proper connection and proper-walk connection numbers in arc-colored digraphs, establishing bounds for circulant digraphs and conditions for equality in Hamiltonian digraphs.

Contribution

It proves that circulant digraphs with certain properties have a proper connection number at most 2 and provides sufficient conditions for Hamiltonian digraphs to have equal proper connection and proper-walk connection numbers.

Findings

Proper connection number of certain circulant digraphs is at most 2.

Conditions identified under which Hamiltonian digraphs have equal proper connection and proper-walk connection numbers.

Theoretical bounds and properties for arc-colored digraph connectivity.

Abstract

An arc-colored digraph D is properly (properly-walk) connected if, for any ordered pair of vertices $(u, v)$, the digraph $D$ contains a directed path (a directed walk) from $u$ to $v$ such that arcs adjacent on that path (on that walk) have distinct colors. The proper connection number $\overrightarrow{pc}(D)$ (the proper-walk connection number $\overrightarrow{wc}(D)$) of a digraph $D$ is the minimum number of colours to make $D$ properly connected (properly-walk connected). We prove that $\overrightarrow{pc}(C_n(S)) \leq 2$ for every circulant digraph $C_n(S)$ with $S\subseteq\{1,\ldots ,n-1\}, |S|\ge 2$ and $1\in S$. Furthermore, we give some sufficient conditions for a Hamiltonian digraph $D$ to satisfy $\overrightarrow{pc}(D)= \overrightarrow{wc}(D) = 2$.