The adjacent vertex distinguishing total coloring of planar graphs without adjacent 4-cycles

A proper [k]-total coloring c of a graph G is a proper total coloring c of G using colors of the set [k] = {1, 2,. .. , k}. Let p(u) denote the product of the color on a vertex u and colors on all the edges incident with u. For each edge uv ∈ E(G), if p(u) = p(v), then we say the coloring c distinguishes adjacent vertices by product and call it a neighbor product distinguishing k-total coloring of G. By χ (G), we denote the smallest value of k in such a coloring of G. It has been conjectured by Li et al. that ∆(G) + 3 colors enable the existence of a neighbor product distinguishing total coloring. In this paper, by applying the Combinatorial Nullstellensatz, we obtain that the conjecture holds for planar graph with ∆(G) ≥ 10. Moreover, for planar graph G with ∆(G) ≥ 11, it is neighbor product distinguishing (∆(G) + 2)total colorable, and the upper bound ∆(G) + 2 is tight.

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The adjacent vertex distinguishing total coloring of planar graphs without adjacent 4-cycles

Cited by 3 publications

References 27 publications

The adjacent vertex distinguishing total choosability of planar graphs with maximum degree at least eleven

The adjacent vertex distinguishing total choosability of planar graphs with maximum degree at least eleven

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Neighbor product distinguishing total colorings of planar graphs with maximum degree at least ten

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