| Title:
|
Neighbor sum distinguishing list total coloring of IC-planar graphs without 5-cycles (English) |
| Author:
|
Zhang, Donghan |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
72 |
| Issue:
|
1 |
| Year:
|
2022 |
| Pages:
|
111-124 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $G=(V(G),E(G))$ be a simple graph and $E_{G}(v)$ denote the set of edges incident with a vertex $v$. A neighbor sum distinguishing (NSD) total coloring $\phi $ of $G$ is a proper total coloring of $G$ such that $\sum _{z\in E_{G}(u)\cup \{u\}}\phi (z)\neq \sum _{z\in E_{G}(v)\cup \{v\}}\phi (z)$ for each edge $uv\in E(G)$. Pilśniak and Woźniak asserted in 2015 that each graph with maximum degree $\Delta $ admits an NSD total $(\Delta +3)$-coloring. We prove that the list version of this conjecture holds for any IC-planar graph with $\Delta \geq 11$ but without $5$-cycles by applying the Combinatorial Nullstellensatz. (English) |
| Keyword:
|
IC-planar graph |
| Keyword:
|
neighbor sum distinguishing list total coloring |
| Keyword:
|
Combinatorial Nullstellensatz |
| Keyword:
|
discharging method |
| MSC:
|
05C10 |
| MSC:
|
05C15 |
| idZBL:
|
Zbl 07511556 |
| idMR:
|
MR4389109 |
| DOI:
|
10.21136/CMJ.2021.0333-20 |
| . |
| Date available:
|
2022-03-25T08:27:23Z |
| Last updated:
|
2024-04-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149576 |
| . |
| Reference:
|
[1] Albertson, M. O.: Chromatic number, independent ratio, and crossing number.Ars Math. Contemp. 1 (2008), 1-6. Zbl 1181.05032, MR 2434266, 10.26493/1855-3974.10.2d0 |
| Reference:
|
[2] Alon, N.: Combinatorial Nullstellensatz.Comb. Probab. Comput. 8 (1999), 7-29. Zbl 0920.05026, MR 1684621, 10.1017/S0963548398003411 |
| Reference:
|
[3] Bondy, J. A., Murty, U. S. R.: Graph Theory.Graduate Texts in Mathematics 244. Springer, Berlin (2008). Zbl 1134.05001, MR 2368647, 10.1007/978-1-84628-970-5 |
| Reference:
|
[4] Pilśniak, M., Woźniak, M.: On the total-neighbor-distinguishing index by sums.Graphs Comb. 31 (2015), 771-782. Zbl 1312.05054, MR 3338032, 10.1007/s00373-013-1399-4 |
| Reference:
|
[5] Qu, C., Wang, G., Yan, G., Yu, X.: Neighbor sum distinguishing total choosability of planar graphs.J. Comb. Optim. 32 (2016), 906-916. Zbl 1348.05082, MR 3544074, 10.1007/s10878-015-9911-9 |
| Reference:
|
[6] Song, C., Jin, X., Xu, C.: Neighbor sum distinguishing total coloring of IC-planar graphs with short cycle restrictions.Discrete Appl. Math. 279 (2020), 202-209. Zbl 1439.05095, MR 4092640, 10.1016/j.dam.2019.12.023 |
| Reference:
|
[7] Song, C., Xu, C.: Neighbor sum distinguishing total colorings of IC-planar graphs with maximum degree 13.J. Comb. Optim. 39 (2020), 293-303. Zbl 1434.05057, MR 4047108, 10.1007/s10878-019-00467-1 |
| Reference:
|
[8] Song, W., Duan, Y., Miao, L.: Neighbor sum distinguishing total coloring of triangle free IC-planar graphs.Acta Math. Sin., Engl. Ser. 36 (2020), 292-304. Zbl 1439.05096, MR 4072704, 10.1007/s10114-020-9189-4 |
| Reference:
|
[9] Song, W., Miao, L., Duan, Y.: Neighbor sum distinguishing total choosability of IC-planar graphs.Discuss. Math., Graph Theory 40 (2020), 331-344. Zbl 1430.05023, MR 4041985, 10.7151/dmgt.2145 |
| Reference:
|
[10] Wang, J., Cai, J., Qiu, B.: Neighbor sum distinguishing total choosability of planar graphs without adjacent triangles.Theor. Comput. Sci. 661 (2017), 1-7. Zbl 1357.05027, MR 3591208, 10.1016/j.tcs.2016.11.003 |
| Reference:
|
[11] Yang, D., Sun, L., Yu, X., Wu, J., Zhou, S.: Neighbor sum distinguishing total chromatic number of planar graphs with maximum degree 10.Appl. Math. Comput. 314 (2017), 456-468. Zbl 1426.05051, MR 3683886, 10.1016/j.amc.2017.06.002 |
| . |
