Neighbor sum distinguishing total coloring of planar graphs without 5-cycles

Abstract Let G be a graph, a proper total coloring ϕ : V ( G ) ∪ E ( G ) → { 1 , 2 , … , k } is called neighbor sum distinguishing if f ( u ) ≠ f ( v ) for each edge u v ∈ E ( G ) , where f ( v ) = ∑ u v ∈ E ( G ) ϕ ( u v ) + ϕ ( v ) , v ∈ V ( G ) . We use χ Σ ″ ( G ) to denote the smallest number k in such a coloring of G . Pilśniak and Woźniak have already conjectured that χ Σ ″ ( G ) ≤ Δ ( G ) + 3 for any simple graph with maximum degree Δ ( G ) . In this paper, we prove that for any planar graph G without 5-cycles, χ Σ ″ ( G ) ≤ max ⁡ { Δ ( G ) + 3 , 10 } .