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A NOTE ON GROUP COLORINGS AND GROUP STRUCTURE.
- Source :
-
SIAM Journal on Discrete Mathematics . 2021, Vol. 35 Issue 4, p2535-2543. 9p. - Publication Year :
- 2021
-
Abstract
- Abelian group colorings were first introduced by Jaeger et al. in [J. Combin. Theory Ser. B, 56 (1992), pp. 165--182] as the dual concept of group connectivity of graphs. For given groups $\Gamma_1$ and $\Gamma_2$ with $|\Gamma_1| = |\Gamma_2|$, the dual version of a problem raised by Jaeger et al. suggests to investigate whether every $\Gamma_1$-colorable graph $G$ is also $\Gamma_2$-colorable. Recently, Husek, Mohelníková, and Šámal [J. Graph Theory, 93 (2019), pp. 317--327] used computer testing to find the first examples of $\mathbb{Z}_4$-connected but not $\mathbb{Z}_2^2$-connected graphs as well as $\mathbb{Z}_2^2$-connected but not $\mathbb{Z}_4$-connected graphs. As their examples are nonplanar, the group coloring problem remains unanswered. Group coloring was extended to non-abelian groups in Li and Lai [Discrete Math., 313 (2013), pp. 101--104]. We introduce a group coloring local structure (defined as a snarl in the paper) and use it to construct infinitely many ordered triples $(G, \Gamma_1, \Gamma_2)$ in which $G$ is a graph and $\Gamma_1$ and $\Gamma_2$ are groups with $|\Gamma_1| = |\Gamma_2|$, such that $G$ is $\Gamma_1$-colorable but not $\Gamma_2$-colorable. [ABSTRACT FROM AUTHOR]
- Subjects :
- *NONABELIAN groups
*GRAPH connectivity
*ABELIAN groups
*GRAPH theory
*GRAPH coloring
Subjects
Details
- Language :
- English
- ISSN :
- 08954801
- Volume :
- 35
- Issue :
- 4
- Database :
- Academic Search Index
- Journal :
- SIAM Journal on Discrete Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- 154646113
- Full Text :
- https://doi.org/10.1137/19M1300546