{"title":"最大阶数至少为 10 的平面图的邻域和区分总可选择性","authors":"Dong-han Zhang, You Lu, Sheng-gui Zhang, Li Zhang","doi":"10.1007/s10255-024-1110-y","DOIUrl":null,"url":null,"abstract":"<div><p>A neighbor sum distinguishing (NSD) total coloring <i>ϕ</i> of <i>G</i> is a proper total coloring of <i>G</i> such that <span>\\(\\sum\\limits_{z \\in {E_G}(u) \\cup \\{u\\}} {\\phi (z) \\ne} \\sum\\limits_{z \\in {E_G}(v) \\cup \\{v\\}} {\\phi (z)} \\)</span> for each edge <i>uv</i> ∈ <i>E</i>(<i>G</i>), where <i>EG</i>(<i>u</i>) is the set of edges incident with a vertex <i>u</i>. In 2015, Pilśniak and Woźniak conjectured that every graph with maximum degree Δ has an NSD total (Δ + 3)-coloring. Recently, Yang et al. proved that the conjecture holds for planar graphs with Δ ≥ 10, and Qu et al. proved that the list version of the conjecture also holds for planar graphs with Δ ≥ 13. In this paper, we improve their results and prove that the list version of the conjecture holds for planar graphs with Δ ≥ 10.</p></div>","PeriodicalId":6951,"journal":{"name":"Acta Mathematicae Applicatae Sinica, English Series","volume":"40 1","pages":"211 - 224"},"PeriodicalIF":0.9000,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Neighbor Sum Distinguishing Total Choosability of Planar Graphs with Maximum Degree at Least 10\",\"authors\":\"Dong-han Zhang, You Lu, Sheng-gui Zhang, Li Zhang\",\"doi\":\"10.1007/s10255-024-1110-y\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>A neighbor sum distinguishing (NSD) total coloring <i>ϕ</i> of <i>G</i> is a proper total coloring of <i>G</i> such that <span>\\\\(\\\\sum\\\\limits_{z \\\\in {E_G}(u) \\\\cup \\\\{u\\\\}} {\\\\phi (z) \\\\ne} \\\\sum\\\\limits_{z \\\\in {E_G}(v) \\\\cup \\\\{v\\\\}} {\\\\phi (z)} \\\\)</span> for each edge <i>uv</i> ∈ <i>E</i>(<i>G</i>), where <i>EG</i>(<i>u</i>) is the set of edges incident with a vertex <i>u</i>. In 2015, Pilśniak and Woźniak conjectured that every graph with maximum degree Δ has an NSD total (Δ + 3)-coloring. Recently, Yang et al. proved that the conjecture holds for planar graphs with Δ ≥ 10, and Qu et al. proved that the list version of the conjecture also holds for planar graphs with Δ ≥ 13. In this paper, we improve their results and prove that the list version of the conjecture holds for planar graphs with Δ ≥ 10.</p></div>\",\"PeriodicalId\":6951,\"journal\":{\"name\":\"Acta Mathematicae Applicatae Sinica, English Series\",\"volume\":\"40 1\",\"pages\":\"211 - 224\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-01-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematicae Applicatae Sinica, English Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10255-024-1110-y\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematicae Applicatae Sinica, English Series","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10255-024-1110-y","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Neighbor Sum Distinguishing Total Choosability of Planar Graphs with Maximum Degree at Least 10
A neighbor sum distinguishing (NSD) total coloring ϕ of G is a proper total coloring of G such that \(\sum\limits_{z \in {E_G}(u) \cup \{u\}} {\phi (z) \ne} \sum\limits_{z \in {E_G}(v) \cup \{v\}} {\phi (z)} \) for each edge uv ∈ E(G), where EG(u) is the set of edges incident with a vertex u. In 2015, Pilśniak and Woźniak conjectured that every graph with maximum degree Δ has an NSD total (Δ + 3)-coloring. Recently, Yang et al. proved that the conjecture holds for planar graphs with Δ ≥ 10, and Qu et al. proved that the list version of the conjecture also holds for planar graphs with Δ ≥ 13. In this paper, we improve their results and prove that the list version of the conjecture holds for planar graphs with Δ ≥ 10.
期刊介绍:
Acta Mathematicae Applicatae Sinica (English Series) is a quarterly journal established by the Chinese Mathematical Society. The journal publishes high quality research papers from all branches of applied mathematics, and particularly welcomes those from partial differential equations, computational mathematics, applied probability, mathematical finance, statistics, dynamical systems, optimization and management science.