Neighbor sum distinguishing total coloring of planar graphs with restrained cycles.

Given a graph G = (V (G) , E (G)) and a proper total k -coloring μ : V (G) ∪ E (G) → { 1 , 2 , ... , k } , we call μ neighbor sum distinguishing total coloring provided h (v) ≠ h (u) for any u v ∈ E (G) where h (v) = μ (v) + ∑ u v ∈ E (G) μ (u v) for any v ∈ V (G). Neighbor sum distinguishing total coloring was first defined by Pilśniak and Woźniak. They conjectured Δ (G) + 3 colors enable any graph G to admit such a coloring. The neighbor sum distinguishing total chromatic number χ Σ ′ ′ is the minimum integer where a graph is needed for this coloring. In this paper, we present two conclusions that χ Σ ′ ′ ≤ Δ (G) + 2 provided there are no 3-cycles adjacent to 4-cycles in a planar graph G with Δ (G) ≥ 8 without cut edges, and χ Σ ′ ′ ≤ Δ (G) + 3 provided there are no 4-cycles intersecting with 6-cycles in a planar graph G with Δ (G) ≥ 7. [ABSTRACT FROM AUTHOR]