Simultaneous coloring of vertices and incidences of outerplanar graphs

A vi-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 vi-simultaneous chromatic number, denoted by chi vi(G), is the smallest integer k such that G has a vi-simultaneous proper k-coloring. In [M. Mozafari-Nia, M. N. Iradmusa, A note on coloring of 33-power of subquartic graphs, Vol. 79, No.3, 2021] vi-simultaneous proper coloring of graphs with maximum degree 4 is investigated and they conjectured that for any graph G with maximum degree increment >= 2, vi-simultaneous proper coloring of G is at most 2 increment + 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 vi-simultaneous chromatic number of any outerplanar graph G is either increment + 2 or increment + 3, where increment is the maximum degree of G.