| Summary: | For a given graph G = (V (G), E(G)), a proper total coloring ϕ: V (G) ∪ E(G) → {1, 2, . . . , k} is neighbor sum distinguishing if f(u) ≠ f(v) for each edge uv ∈ E(G), where f(v) = Σuv∈E(G) ϕ(uv)+ϕ(v), v ∈ V (G). The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by
χΣ″(G)\chi _\Sigma ^{''} ( G )
. Pilśniak and Woźniak first introduced this coloring and conjectured that
χΣ″(G)≤Δ(G)+3\chi _\Sigma ^{''} ( G ) \le \Delta ( G ) + 3
for any graph with maximum degree Δ(G). In this paper, by using the discharging method, we prove that for any planar graph G without 5-cycles,
χΣ″(G)≤max{Δ(G)+2, 10}\chi _\Sigma ^{''} ( G ) \le \max \left\{ {\Delta ( G ) + 2,\;10} \right\}
. The bound Δ(G) + 2 is sharp. Furthermore, we get the exact value of
χΣ″(G)\chi _\Sigma ^{''} ( G )
if Δ(G) ≥ 9.
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