Neighbor sum distinguishing total choosability of planar graphs without 4-cycles adjacent to 3-cycles

Kittikorn Nakprasit, Patcharapan Jumnongnit

Abstract


Let φ be a proper total coloring of a graph G with integers as colors. For a vertex v, let w(v) denote the sum of colors assigned to edges incident to v and the color assigned to v. If w(u)≠w(v) whenever uv∈E(G), then φ is called a neighbor sum distinguishing total coloring. A k-assignment L of G is a list assignment L of integers to vertices and edges with |L(z)|=k for each z∈V(G)∪E(G). A total-L-coloring is a total coloring φ of G such that φ(v)∈L(v) whenever v∈V(G) and φ(e)∈L(e) whenever e∈E(G). The smallest integer k such that G has a neighbor sum distinguishing total-L-coloring for every k-assignment L is called the neighbor sum distinguishing total choosability of G and is denoted by Ch’’(G). Wang, Cai, and Ma [15] proved that every planar graph G without 4-cycles with ∆(G)≥7 has Ch’’(G)≤∆(G)+3. In this work, we strengthen the result of Wang et al by proving that Ch’’(G)≤∆(G)+3 for every planar graph G without 4-cycles adjacent to 3-cycles with ∆(G)≥7.

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Published: 2022-03-14

How to Cite this Article:

Kittikorn Nakprasit, Patcharapan Jumnongnit, Neighbor sum distinguishing total choosability of planar graphs without 4-cycles adjacent to 3-cycles, J. Math. Comput. Sci., 12 (2022), Article ID 111

Copyright © 2022 Kittikorn Nakprasit, Patcharapan Jumnongnit. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

 

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