{"title":"论图形半强积的阿隆-塔西数","authors":"Lin Niu, Xiangwen Li","doi":"10.1007/s10878-023-01099-2","DOIUrl":null,"url":null,"abstract":"<p>The Alon–Tarsi number was defined by Jensen and Toft (Graph coloring problems, Wiley, New York, 1995). The <i>Alon–Tarsi</i> number <i>AT</i>(<i>G</i>) of a graph <i>G</i> is the smallest integer <i>k</i> such that <i>G</i> has an orientation <i>D</i> with maximum outdegree <span>\\(k-1\\)</span> and the number of even circulation is not equal to that of odd circulations in <i>D</i>. It is known that <span>\\(\\chi (G)\\le \\chi _l(G)\\le AT(G)\\)</span> for any graph <i>G</i>, where <span>\\(\\chi (G)\\)</span> and <span>\\(\\chi _l(G)\\)</span> are the chromatic number and the list chromatic number of <i>G</i>. Denote by <span>\\(H_1 \\square H_2\\)</span> and <span>\\(H_1\\bowtie H_2\\)</span> the Cartesian product and the semi-strong product of two graphs <span>\\(H_1\\)</span> and <span>\\(H_2\\)</span>, respectively. Kaul and Mudrock (Electron J Combin 26(1):P1.3, 2019) proved that <span>\\(AT(C_{2k+1}\\square P_n)=3\\)</span>. Li, Shao, Petrov and Gordeev (Eur J Combin 103697, 2023) proved that <span>\\(AT(C_n\\square C_{2k})=3\\)</span> and <span>\\(AT(C_{2m+1}\\square C_{2n+1})=4\\)</span>. Petrov and Gordeev (Mosc. J. Comb. Number Theory 10(4):271–279, 2022) proved that <span>\\(AT(K_n\\square C_{2k})=n\\)</span>. Note that the semi-strong product is noncommutative. In this paper, we determine <span>\\(AT(P_m \\bowtie P_n)\\)</span>, <span>\\(AT(C_m \\bowtie C_{2n})\\)</span>, <span>\\(AT(C_m \\bowtie P_n)\\)</span> and <span>\\(AT(P_m \\bowtie C_{n})\\)</span>. We also prove that <span>\\(5\\le AT(C_m \\bowtie C_{2n+1})\\le 6\\)</span>.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the Alon–Tarsi number of semi-strong product of graphs\",\"authors\":\"Lin Niu, Xiangwen Li\",\"doi\":\"10.1007/s10878-023-01099-2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The Alon–Tarsi number was defined by Jensen and Toft (Graph coloring problems, Wiley, New York, 1995). The <i>Alon–Tarsi</i> number <i>AT</i>(<i>G</i>) of a graph <i>G</i> is the smallest integer <i>k</i> such that <i>G</i> has an orientation <i>D</i> with maximum outdegree <span>\\\\(k-1\\\\)</span> and the number of even circulation is not equal to that of odd circulations in <i>D</i>. It is known that <span>\\\\(\\\\chi (G)\\\\le \\\\chi _l(G)\\\\le AT(G)\\\\)</span> for any graph <i>G</i>, where <span>\\\\(\\\\chi (G)\\\\)</span> and <span>\\\\(\\\\chi _l(G)\\\\)</span> are the chromatic number and the list chromatic number of <i>G</i>. Denote by <span>\\\\(H_1 \\\\square H_2\\\\)</span> and <span>\\\\(H_1\\\\bowtie H_2\\\\)</span> the Cartesian product and the semi-strong product of two graphs <span>\\\\(H_1\\\\)</span> and <span>\\\\(H_2\\\\)</span>, respectively. Kaul and Mudrock (Electron J Combin 26(1):P1.3, 2019) proved that <span>\\\\(AT(C_{2k+1}\\\\square P_n)=3\\\\)</span>. Li, Shao, Petrov and Gordeev (Eur J Combin 103697, 2023) proved that <span>\\\\(AT(C_n\\\\square C_{2k})=3\\\\)</span> and <span>\\\\(AT(C_{2m+1}\\\\square C_{2n+1})=4\\\\)</span>. Petrov and Gordeev (Mosc. J. Comb. Number Theory 10(4):271–279, 2022) proved that <span>\\\\(AT(K_n\\\\square C_{2k})=n\\\\)</span>. Note that the semi-strong product is noncommutative. In this paper, we determine <span>\\\\(AT(P_m \\\\bowtie P_n)\\\\)</span>, <span>\\\\(AT(C_m \\\\bowtie C_{2n})\\\\)</span>, <span>\\\\(AT(C_m \\\\bowtie P_n)\\\\)</span> and <span>\\\\(AT(P_m \\\\bowtie C_{n})\\\\)</span>. We also prove that <span>\\\\(5\\\\le AT(C_m \\\\bowtie C_{2n+1})\\\\le 6\\\\)</span>.</p>\",\"PeriodicalId\":50231,\"journal\":{\"name\":\"Journal of Combinatorial Optimization\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-01-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Optimization\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10878-023-01099-2\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Optimization","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10878-023-01099-2","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
On the Alon–Tarsi number of semi-strong product of graphs
The Alon–Tarsi number was defined by Jensen and Toft (Graph coloring problems, Wiley, New York, 1995). The Alon–Tarsi number AT(G) of a graph G is the smallest integer k such that G has an orientation D with maximum outdegree \(k-1\) and the number of even circulation is not equal to that of odd circulations in D. It is known that \(\chi (G)\le \chi _l(G)\le AT(G)\) for any graph G, where \(\chi (G)\) and \(\chi _l(G)\) are the chromatic number and the list chromatic number of G. Denote by \(H_1 \square H_2\) and \(H_1\bowtie H_2\) the Cartesian product and the semi-strong product of two graphs \(H_1\) and \(H_2\), respectively. Kaul and Mudrock (Electron J Combin 26(1):P1.3, 2019) proved that \(AT(C_{2k+1}\square P_n)=3\). Li, Shao, Petrov and Gordeev (Eur J Combin 103697, 2023) proved that \(AT(C_n\square C_{2k})=3\) and \(AT(C_{2m+1}\square C_{2n+1})=4\). Petrov and Gordeev (Mosc. J. Comb. Number Theory 10(4):271–279, 2022) proved that \(AT(K_n\square C_{2k})=n\). Note that the semi-strong product is noncommutative. In this paper, we determine \(AT(P_m \bowtie P_n)\), \(AT(C_m \bowtie C_{2n})\), \(AT(C_m \bowtie P_n)\) and \(AT(P_m \bowtie C_{n})\). We also prove that \(5\le AT(C_m \bowtie C_{2n+1})\le 6\).
期刊介绍:
The objective of Journal of Combinatorial Optimization is to advance and promote the theory and applications of combinatorial optimization, which is an area of research at the intersection of applied mathematics, computer science, and operations research and which overlaps with many other areas such as computation complexity, computational biology, VLSI design, communication networks, and management science. It includes complexity analysis and algorithm design for combinatorial optimization problems, numerical experiments and problem discovery with applications in science and engineering.
The Journal of Combinatorial Optimization publishes refereed papers dealing with all theoretical, computational and applied aspects of combinatorial optimization. It also publishes reviews of appropriate books and special issues of journals.