Symmetric Set Coloring of Signed Graphs

C. Cappello, E. Steffen, Annals of Combinatorics (2022).

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Journal Article | Published | English
Abstract
There are many concepts of signed graph coloring which are defined by assigning colors to the vertices of the graphs. These concepts usually differ in the number of self-inverse colors used. We introduce a unifying concept for this kind of coloring by assigning elements from symmetric sets to the vertices of the signed graphs. In the first part of the paper, we study colorings with elements from symmetric sets where the number of self-inverse elements is fixed. We prove a Brooks’-type theorem and upper bounds for the corresponding chromatic numbers in terms of the chromatic number of the underlying graph. These results are used in the second part where we introduce the symset-chromatic number χsym(G,σ) of a signed graph (G,σ). We show that the symset-chromatic number gives the minimum partition of a signed graph into independent sets and non-bipartite antibalanced subgraphs. In particular, χsym(G,σ)≤χ(G). In the final section we show that these colorings can also be formalized as DP-colorings.
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Annals of Combinatorics
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Cappello C, Steffen E. Symmetric Set Coloring of Signed Graphs. Annals of Combinatorics. Published online 2022. doi:10.1007/s00026-022-00593-4
Cappello, C., & Steffen, E. (2022). Symmetric Set Coloring of Signed Graphs. Annals of Combinatorics. https://doi.org/10.1007/s00026-022-00593-4
@article{Cappello_Steffen_2022, title={Symmetric Set Coloring of Signed Graphs}, DOI={10.1007/s00026-022-00593-4}, journal={Annals of Combinatorics}, publisher={Springer Science and Business Media LLC}, author={Cappello, Chiara and Steffen, Eckhard}, year={2022} }
Cappello, Chiara, and Eckhard Steffen. “Symmetric Set Coloring of Signed Graphs.” Annals of Combinatorics, 2022. https://doi.org/10.1007/s00026-022-00593-4.
C. Cappello and E. Steffen, “Symmetric Set Coloring of Signed Graphs,” Annals of Combinatorics, 2022, doi: 10.1007/s00026-022-00593-4.
Cappello, Chiara, and Eckhard Steffen. “Symmetric Set Coloring of Signed Graphs.” Annals of Combinatorics, Springer Science and Business Media LLC, 2022, doi:10.1007/s00026-022-00593-4.

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