On vertex in-out-antimagic total digraphs (Q6552596)

The concept of a vertex in-out-antimagic total labeling tightens the requirements of previously introduced labelings in [\textit{S. Arumugam} et al., J. Discrete Math. Sci. Cryptography 25, No. 6, 1603--1611 (2022; Zbl 1496.05156); \textit{G. Marimuthu} et al., J. Graph Label. 1, No. 1, 21--30 (2015; Zbl 1334.05149)]. The natural question is whether digraphs exist with total labeling that are simultaneously vertex in-antimagic and vertex out-antimagic. In this paper, the authors give a partial answer to this question. The aim is to create a total labeling that is both out-antimagic and in-antimagic. The authors conjecture that all digraphs allow such labeling and provide certain general classes of digraphs that support this conjecture. In Section 2, they present a basic formulation for relatively dense digraphs: the smallest indegree or the smallest outdegree exceeds \(\sqrt{n-1}\), where \(n\) is the number of vertices of the digraph. Section 3 provides a construction for sparse digraphs, and Section 4 includes constructive proofs of vertex in-out-antimagic total labelings for several other infinite classes of digraphs. However, the main conjecture ``All digraphs allow a vertex in-out-antimagic total labeling'' remains unresolved. They believe that a new construction of a vertex in-out-antimagic total labeling covering all digraphs with outdegree 1 would not only be a solution to a missing regularity not covered by the theorem ``Let \(D\) be a digraph with \(n\) vertices with every vertex of the same outdegree or the same indegree \(r\) greater than 1. Then, the digraph \(D\) allows a vertex in-out-antimagic total labeling,'' but also possibly leads to a solution of the conjecture ``All digraphs allow a vertex in-out-antimagic total labeling''.\N\NThis paper is quite well-written. Reading this paper will be really helpful to researchers. This study will provide researchers with insight into graph labeling studies.