Simultaneous coloring of vertices and incidences of Outerplanar graphs
Mahsa Mozafari-Nia, Moharram N. Iradmusa

TL;DR
This paper investigates the $vi$-simultaneous proper coloring of outerplanar graphs, establishing that their chromatic number is either $ ext{max degree}+2$ or $ ext{max degree}+3$, advancing understanding of coloring properties in graph theory.
Contribution
It proves that the $vi$-simultaneous chromatic number for outerplanar graphs is tightly bounded between $ ext{max degree}+2$ and $ ext{max degree}+3$, providing a precise characterization.
Findings
The $vi$-simultaneous chromatic number of outerplanar graphs is at most $ ext{max degree}+3$.
The chromatic number is at least $ ext{max degree}+2$ for these graphs.
The exact value is either $ ext{max degree}+2$ or $ ext{max degree}+3$.
Abstract
A -simultaneous proper -coloring of a graph is a coloring of all vertices and incidences of the graph in which any two adjacent or incident elements in the set receive distinct colors, where is the set of incidences of . The -simultaneous chromatic number, denoted by , is the smallest integer such that has a -simultaneous proper -coloring. In [M. Mozafari-Nia, M. N. Iradmusa, A note on coloring of -power of subquartic graphs, Vol. 79, No.3, 2021] -simultaneous proper coloring of graphs with maximum degree is investigated and they conjectured that for any graph with maximum degree , -simultaneous proper coloring of is at most . In [M. Mozafari-Nia, M. N. Iradmusa, Simultaneous coloring of vertices and incidences of graphs, arXiv:2205.07189, 2022] the correctness…
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Taxonomy
TopicsNuclear Receptors and Signaling
