Sylwia Cichacz ; Jakub Przybyło - Group twin coloring of graphs

dmtcs:3948 - Discrete Mathematics & Theoretical Computer Science, June 26, 2018, Vol. 20 no. 1 -
Group twin coloring of graphsArticle

Authors: Sylwia Cichacz ; Jakub Przybyło

    For a given graph $G$, the least integer $k\geq 2$ such that for every Abelian group $\mathcal{G}$ of order $k$ there exists a proper edge labeling $f:E(G)\rightarrow \mathcal{G}$ so that $\sum_{x\in N(u)}f(xu)\neq \sum_{x\in N(v)}f(xv)$ for each edge $uv\in E(G)$ is called the \textit{group twin chromatic index} of $G$ and denoted by $\chi'_g(G)$. This graph invariant is related to a few well-known problems in the field of neighbor distinguishing graph colorings. We conjecture that $\chi'_g(G)\leq \Delta(G)+3$ for all graphs without isolated edges, where $\Delta(G)$ is the maximum degree of $G$, and provide an infinite family of connected graph (trees) for which the equality holds. We prove that this conjecture is valid for all trees, and then apply this result as the base case for proving a general upper bound for all graphs $G$ without isolated edges: $\chi'_g(G)\leq 2(\Delta(G)+{\rm col}(G))-5$, where ${\rm col}(G)$ denotes the coloring number of $G$. This improves the best known upper bound known previously only for the case of cyclic groups $\mathbb{Z}_k$.

    Volume: Vol. 20 no. 1
    Section: Graph Theory
    Published on: June 26, 2018
    Accepted on: June 5, 2018
    Submitted on: September 22, 2017
    Keywords: Mathematics - Combinatorics

    Consultation statistics

    This page has been seen 466 times.
    This article's PDF has been downloaded 420 times.