Quasi-majority neighbor sum distinguishing edge-colorings
Rafał Kalinowski, Monika Pilśniak, Elżbieta Sidorowicz, Elżbieta Turowska
TL;DR
The paper studies edge-colorings that induce a neighbor-sum distinguishing vertex coloring under a quasi-majority constraint, introducing the QM-NSD index χ_sum^{QM}(G) for nice graphs. It develops both general and class-specific bounds, proving a universal 12-color upper bound and tighter results for bipartite graphs and Δ≤4 graphs, while obtaining exact values for complete graphs, complete bipartite graphs, and trees, along with a majority-NSD variant. The approach combines constructive colorings, interval-coloring insights, and the Combinatorial Nullstellensatz to guarantee the quasi-majority property and neighbor-sum distinctions. These results advance our understanding of constrained edge-colorings with sums and have potential implications for labeling theory and related combinatorial optimization problems.
Abstract
In this paper, a $k$-edge-coloring of $G$ is any mapping $c:E(G)\longrightarrow [k]$. The edge-coloring $c$ of $G$ naturally defines a vertex-coloring $σ_{c}: V(G) \to \mathbb{N}$, where $σ_{c}(v)=\sum_{u\in N_G(v)}c(vu)$ for every vertex $v\in V(G)$. The edge-coloring $c$ is said to be neighbor sum distinguishing if it results in a proper vertex-coloring $σ_{c}$, which that $σ_{c}(u) \neq σ_{c}(v)$ for every edge $uv$ in $G$. We investigate neighbor sum distinguishing edge-colorings with local constraints, where the edge-coloring is quasi-majority at each vertex. Specifically, every vertex $v$ is incident to at most $\left\lceil d(v)/2 \right\rceil$ edges of one color. This type of coloring is referred to as quasi-majority neighbor sum distinguishing edge-coloring. The minimum number of colors required for a graph to have a quasi-majority neighbor sum distinguishing edge-coloring is called the quasi-majority neighbor sum distinguishing index. A graph is nice if it has no component isomorphic to $K_2$. We prove that any nice graph admits a quasi-majority neighbor sum distinguishing edge-coloring using at most 12 colors. This bound can be improved for bipartite graphs and graphs with a maximum degree of at most 4. Specifically, we show that every nice bipartite graph can be colored with 6 colors, and every nice graph with a maximum degree of at most 4 can be colored with 7 colors. Additionally, we determine the exact value of the quasi-majority neighbor sum distinguishing index for complete graphs, complete bipartite graphs, and trees. We also consider majority neighbor sum distinguishing edge-colorings, that is, when each vertex is incident to at most $d(v)/2$ edges with the same color.