### Simultaneous coloring of vertices and incidences of outerplanar graphs

#### Abstract

A *v**i*-simultaneous proper *k*-coloring of a graph *G* is a coloring of all vertices and incidences of the graph in which any two adjacent or incident elements in the set *V*(*G*)∪*I*(*G*) receive distinct colors, where *I*(*G*) is the set of incidences of *G*. The *v**i*-simultaneous chromatic number, denoted by *χ*_{vi}(*G*), is the smallest integer *k* such that *G* has a *v**i*-simultaneous proper *k*-coloring. In [M. Mozafari-Nia, M. N. Iradmusa, A note on coloring of 3/3-power of subquartic graphs, Vol. 79, No.3, 2021] *v**i*-simultaneous proper coloring of graphs with maximum degree 4 is investigated and they conjectured that for any graph *G* with maximum degree *Δ* ≥ 2, *v**i*-simultaneous proper coloring of *G* is at most 2*Δ* + 1. In [M. Mozafari-Nia, M. N. Iradmusa, Simultaneous coloring of vertices and incidences of graphs, arXiv:2205.07189, 2022] the correctness of the conjecture for some classes of graphs such as *k*-degenerated graphs, cycles, forests, complete graphs, regular bipartite graphs is investigated. In this paper, we prove that the *v**i*-simultaneous chromatic number of any outerplanar graph *G* is either *Δ* + 2 or *Δ* + 3, where *Δ* is the maximum degree of *G*.

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PDFDOI: http://dx.doi.org/10.5614/ejgta.2023.11.1.20

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