{"title":"Minor classes (extended abstract)","authors":"D. Vertigan","doi":"10.1090/conm/147/01195","DOIUrl":"https://doi.org/10.1090/conm/147/01195","url":null,"abstract":"Gimbel and Thomassen asked whether 3-colorability of a triangle-free graph drawn on a fixed surface can be tested in polynomial time. We settle the question by giving a linear-time algorithm for every surface which combined with previous results gives a lineartime algorithm to compute the chromatic number of such graphs. Our algorithm is based on a structure theorem that for a triangle-free graph drawn on a surface Σ guarantees the existence of a subgraph H, whose size depends only on Σ, such that there is an easy test whether a 3-coloring of H extends to a 3-coloring of G. The test is based on a topological obstruction, called the “winding number” of a 3-coloring. To prove the structure theorem we make use of disjoint paths with specified ends to find a 3-coloring. If the input triangle-free graph G drawn in Σ is 3colorable we can find a 3-coloring in quadratic time, and if G quadrangulates Σ then we can find the 3coloring in linear time. The latter algorithm requires two ingredients that may be of independent interest: a generalization of a data structure of Kowalik and Kurowski to weighted graphs and a speedup of a disjoint paths algorithm of Robertson and Seymour to linear time.","PeriodicalId":381076,"journal":{"name":"Graph Structure Theory","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125490846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Graph planarity and related topics","authors":"A. Kelmans","doi":"10.1090/conm/147/01205","DOIUrl":"https://doi.org/10.1090/conm/147/01205","url":null,"abstract":"","PeriodicalId":381076,"journal":{"name":"Graph Structure Theory","volume":"97 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115576549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Intercyclic digraphs","authors":"W. McCuaig","doi":"10.1090/conm/147/01176","DOIUrl":"https://doi.org/10.1090/conm/147/01176","url":null,"abstract":"Paths, cycles, trees and sub(di)graphs in directed graphs – p. 1/53","PeriodicalId":381076,"journal":{"name":"Graph Structure Theory","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121269600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Subexponentially computable truncations of Jones-types polynomials","authors":"T. Przytycka, J. Przytycki","doi":"10.1090/conm/147/01165","DOIUrl":"https://doi.org/10.1090/conm/147/01165","url":null,"abstract":"We show that an essential part of the new (Jones-type) polynomial link invariants can be computed in subexponential time. This is in a sharp contrast to the result of Jaeger, Vertigan and Welsh that computing the whole polynomial and most of its evaluations is #P-hard.","PeriodicalId":381076,"journal":{"name":"Graph Structure Theory","volume":"182 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124604548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On a new graph invariant and a criterion for planarity","authors":"Y. D. Verdière","doi":"10.1090/conm/147/01168","DOIUrl":"https://doi.org/10.1090/conm/147/01168","url":null,"abstract":"[9] R. Bacher and Y. Colin deVerdì ere: Multiplicités des valeurs propres et transformationsétoile-triangle des graphes, Bull.","PeriodicalId":381076,"journal":{"name":"Graph Structure Theory","volume":"111 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133845536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Four problems on plane graphs raised by Branko Grünbaum","authors":"O. Borodin","doi":"10.1090/conm/147/01169","DOIUrl":"https://doi.org/10.1090/conm/147/01169","url":null,"abstract":"","PeriodicalId":381076,"journal":{"name":"Graph Structure Theory","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121628136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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