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- Title
The distinguishing index of graphs with infinite minimum degree.
- Authors
Stawiski, Marcin; Wilson, Trevor M.
- Abstract
The distinguishing index D′(G) $D^{\prime} (G)$ of a graph G $G$ is the least number of colors necessary to obtain an edge coloring of G $G$ that is preserved only by the trivial automorphism. We show that if G $G$ is a connected α $\alpha $‐regular graph for some infinite cardinal α $\alpha $ then D′(G)≤2 $D^{\prime} (G)\le 2$, proving a conjecture of Lehner, Pilśniak, and Stawiski. We also show that if G $G$ is a graph with infinite minimum degree and at most 2α ${2}^{\alpha }$ vertices of degree α $\alpha $ for every infinite cardinal α $\alpha $, then D′(G)≤3 $D^{\prime} (G)\le 3$. In particular, D′(G)≤3 $D^{\prime} (G)\le 3$ if G $G$ has infinite minimum degree and order at most 2ℵ0 ${2}^{{\aleph }_{0}}$.
- Subjects
GRAPH coloring; REGULAR graphs; COLORS; LOGICAL prediction
- Publication
Journal of Graph Theory, 2024, Vol 105, Issue 1, p61
- ISSN
0364-9024
- Publication type
Article
- DOI
10.1002/jgt.23013