{"title":"图的迭代Mycielski图的L(2,1)-标记及与匹配问题有关的一些问题","authors":"Kamal Dliou, H. El Boujaoui, M. Kchikech","doi":"10.7151/dmgt.2457","DOIUrl":null,"url":null,"abstract":"Abstract In this paper, we study the L(2, 1)-labeling of the Mycielski graph and the iterated Mycielski graph of graphs in general. For a graph G and all t ≥ 1, we give sharp bounds for λ(Mt(G)) the L(2, 1)-labeling number of the t-th iterated Mycielski graph in terms of the number of iterations t, the order n of G, the maximum degree Δ, and λ(G) the L(2, 1)-labeling number of G. For t = 1, we present necessary and sufficient conditions between the 4-star matching number of the complement graph and λ(M(G)) the L(2, 1)-labeling number of the Mycielski graph of a graph, with some applications to special graphs. For all t ≥ 2, we prove that for any graph G of order n, we have 2t−1(n + 2) − 2 ≤ λ(Mt(G)) ≤ 2t(n + 1) − 2. Thereafter, we characterize the graphs achieving the upper bound 2t(n+1)−2, then by using the Marriage Theorem and Tutte’s characterization of graphs with a perfect 2-matching, we characterize all graphs without isolated vertices achieving the lower bound 2t−1(n + 2) − 2. We determine the L(2, 1)-labeling number for the Mycielski graph and the iterated Mycielski graph of some graph classes.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2022-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"L(2, 1)-Labeling of the Iterated Mycielski Graphs of Graphs and Some Problems Related to Matching Problems\",\"authors\":\"Kamal Dliou, H. El Boujaoui, M. Kchikech\",\"doi\":\"10.7151/dmgt.2457\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract In this paper, we study the L(2, 1)-labeling of the Mycielski graph and the iterated Mycielski graph of graphs in general. For a graph G and all t ≥ 1, we give sharp bounds for λ(Mt(G)) the L(2, 1)-labeling number of the t-th iterated Mycielski graph in terms of the number of iterations t, the order n of G, the maximum degree Δ, and λ(G) the L(2, 1)-labeling number of G. For t = 1, we present necessary and sufficient conditions between the 4-star matching number of the complement graph and λ(M(G)) the L(2, 1)-labeling number of the Mycielski graph of a graph, with some applications to special graphs. For all t ≥ 2, we prove that for any graph G of order n, we have 2t−1(n + 2) − 2 ≤ λ(Mt(G)) ≤ 2t(n + 1) − 2. Thereafter, we characterize the graphs achieving the upper bound 2t(n+1)−2, then by using the Marriage Theorem and Tutte’s characterization of graphs with a perfect 2-matching, we characterize all graphs without isolated vertices achieving the lower bound 2t−1(n + 2) − 2. We determine the L(2, 1)-labeling number for the Mycielski graph and the iterated Mycielski graph of some graph classes.\",\"PeriodicalId\":48875,\"journal\":{\"name\":\"Discussiones Mathematicae Graph Theory\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2022-05-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discussiones Mathematicae Graph Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.7151/dmgt.2457\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discussiones Mathematicae Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.7151/dmgt.2457","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
L(2, 1)-Labeling of the Iterated Mycielski Graphs of Graphs and Some Problems Related to Matching Problems
Abstract In this paper, we study the L(2, 1)-labeling of the Mycielski graph and the iterated Mycielski graph of graphs in general. For a graph G and all t ≥ 1, we give sharp bounds for λ(Mt(G)) the L(2, 1)-labeling number of the t-th iterated Mycielski graph in terms of the number of iterations t, the order n of G, the maximum degree Δ, and λ(G) the L(2, 1)-labeling number of G. For t = 1, we present necessary and sufficient conditions between the 4-star matching number of the complement graph and λ(M(G)) the L(2, 1)-labeling number of the Mycielski graph of a graph, with some applications to special graphs. For all t ≥ 2, we prove that for any graph G of order n, we have 2t−1(n + 2) − 2 ≤ λ(Mt(G)) ≤ 2t(n + 1) − 2. Thereafter, we characterize the graphs achieving the upper bound 2t(n+1)−2, then by using the Marriage Theorem and Tutte’s characterization of graphs with a perfect 2-matching, we characterize all graphs without isolated vertices achieving the lower bound 2t−1(n + 2) − 2. We determine the L(2, 1)-labeling number for the Mycielski graph and the iterated Mycielski graph of some graph classes.
期刊介绍:
The Discussiones Mathematicae Graph Theory publishes high-quality refereed original papers. Occasionally, very authoritative expository survey articles and notes of exceptional value can be published. The journal is mainly devoted to the following topics in Graph Theory: colourings, partitions (general colourings), hereditary properties, independence and domination, structures in graphs (sets, paths, cycles, etc.), local properties, products of graphs as well as graph algorithms related to these topics.