Distance Labelings and (Total-)Neighbor-Distinguishing Colorings of the Edge-Multiplicity-Paths-Replacements

Abstract

As a promotion of the channel assignment problem, an \(L(1,1,1)\)-labeling of a graph \(G\) is an assignment of nonnegative integers to \(V(G)\) such that the difference between labels of adjacent vertices is at least \(1\), and the difference between labels of vertices that are distance two and three apart is at least \(1\). About \(10\) years ago, many mathematicians considered colorings (proper, general, total or from lists) such that vertices (all or adjacent) are distinguished either by sets or multisets or sums. In this paper, we will study \(L(1,1,1)\)-labeling-number and \(L(1,1)\)-edge-labeling-number of the edge-path-replacement. From this, we will consider the total-neighbor-distinguishing coloring and the neighbor-distinguishing coloring of the edge-multiplicity-paths-replacements, give a reference for the conjectures: \(\text{tndis-}_\Sigma(G) \leq \Delta + 3\), \(\text{ndi}_\Sigma(G) \leq \Delta + 2\), and \(\text{tndi}_S(G) \leq \Delta + 3\) for the edge-multiplicity-paths-replacements \(G(rP_k)\) with \(k \geq 3\) and \(r \geq 1\).