Weak and strong local irregularity of digraphs

TL;DR

The paper extends the theory of local irregularity from undirected graphs to digraphs by introducing two new notions: weak local irregularity, based on distinct indegree/outdegree pairs $(d^+(x),d^-(x))$ for adjacent vertices, and strong local irregularity, based on distinct balanced degrees $\sigma(x)=d^+(x)-d^-(x)$. It defines the corresponding coloring indices $\mathrm{lir}(D)$ and $\mathrm{slir}(D)$ and investigates their behavior across digraph classes via skeleton analysis, orientations, and special families like tournaments and cacti. Key results include tight or near-tight bounds for $\mathrm{lir}(D)$ in bipartite, tripartite, planar, complete, and dense-skeleton cases, as well as for tournaments; and significant progress toward a universal bound for $\mathrm{slir}(D)$, with $\mathrm{slir}(D)\le 2$ for several important families (symmetric digraphs, cacti, unicyclic graphs, Eulerian digraphs) and a general bound $\mathrm{slir}(D)\le 3\lceil \log_3 k\rceil$ in terms of the skeleton’s chromatic number $k$. The work also advances conjectures that $\mathrm{lir}(D)\le 2$ and $\mathrm{slir}(D)\le 2$ hold broadly, and it highlights promising directions for constant-bounded digraph colorings and further structural decompositions.

Abstract

Local Irregularity Conjecture states that every simple connected graph, except special cacti, can be decomposed into at most three locally irregular graphs, i.e., graphs in which adjacent vertices have different degrees. The connected minimization problem, finding the minimum number $k$ such that a graph can be decomposed into $k$ locally irregular graphs, is known to be NP-hard in general (Baudon, Bensmail, and Sopena, 2015). This naturally raises interest in the study of related problems. Among others, the concept of local irregularity was defined for digraphs in several different ways. In this paper we present the following new methods of defining a locally irregular digraph. The first one, weak local irregularity, is based on distinguishing adjacent vertices by indegree-outdegree pairs, and the second one, strong local irregularity, asks for different balanced degrees (i.e., difference between the outdegree and the indegree of a vertex) of adjacent vertices. For both of these irregularities, we define locally irregular decompositions and colorings of digraphs. We discuss relation of these concept to others, which were studied previously, and provide related conjectures on the minimum number of colors in weak and strong locally irregular colorings. We support these conjectures with new results, using the chromatic and structural properties of digraphs and their skeletons (Eulerian and symmetric digraphs, orientations of regular graphs, cacti, etc.).

Figures (4)

• Figure 1: Locally irregular coloring of the bow-tie graph $B$ using four colors.
• Figure 2: Scheme of a locally irregular decomposition of a digraph whose skeleton is 6-partite.
• Figure 3: Four cases for which a solution was not found using a computer.
• Figure 4: Cases considered in Claim \ref{['claim5']}.

Theorems & Definitions (57)

• Conjecture 1: Local Irregularity Conjecture Baudon Bensmail Przybylo Wozniak, Sedlar Skrekovski 2
• Conjecture 2: Bensmail Renault
• Conjecture 3: Bensmail Filasto Hocquard Marcille
• Conjecture 4
• Theorem 5
• Theorem 6
• proof
• Theorem 7
• proof
• Theorem 8