Neighbor sum distinguishing total coloring of planar graphs with restrained cycles

Abstract

Given a graph [Formula: see text] and a proper total [Formula: see text]-coloring [Formula: see text]: [Formula: see text], we call [Formula: see text] neighbor sum distinguishing total coloring provided [Formula: see text] for any [Formula: see text] where [Formula: see text] for any [Formula: see text]. Neighbor sum distinguishing total coloring was first defined by Pilśniak and Woźniak. They conjectured [Formula: see text] colors enable any graph [Formula: see text] to admit such a coloring. The neighbor sum distinguishing total chromatic number [Formula: see text] is the minimum integer where a graph is needed for this coloring. In this paper, we present two conclusions that [Formula: see text] provided there are no 3-cycles adjacent to 4-cycles in a planar graph [Formula: see text] with [Formula: see text] without cut edges, and [Formula: see text] provided there are no 4-cycles intersecting with 6-cycles in a planar graph [Formula: see text] with [Formula: see text].