{"title":"正则I类图的流量指数","authors":"Jiaao Li, Xueliang Li, Meiling Wang","doi":"10.1137/21m1393169","DOIUrl":null,"url":null,"abstract":"For integers k and d with k ≥ 2 d > 0, a circular k/d -flow of a graph G is an orientation together with a mapping from E ( G ) to {± d, ± ( d + 1) , . . . , ± ( k − d ) } such that, for each vertex of G , the sum of images on outgoing edges is equal to the sum of images on incoming edges. Related to the Four Color Problem, a classical result of Tutte shows that a cubic graph admits a circular 4 / 1-flow if and only if it is Class I (i.e., 3-edge-colorable). Tutte’s 3-flow conjecture implies that every 5-regular Class I graph admits a nowhere-zero 3-flow (equivalently, a circular 6 / 2-flow) as a special case. Steffen in 2015 conjectured that every (2 t + 1)-regular Class I graph admits a circular (2 t + 2) /t -flow. He also proposed a more general conjecture that every (2 t + 1)-odd-edge-connected (2 t + 1)-regular graph admits a circular (2 t + 2) /t -flow for any integer t ≥ 2, which includes the Circular Flow Conjecture of Jaeger(1981) stating that every 2 t -edge-connected graph admits a circular (2 t + 2) /t -flow for any even t ≥ 2. Jaeger’s conjecture was disproved in 2018 for all even t ≥ 6, and based on these results, Mattiolo and Steffen recently constructed counterexamples to Steffen’s conjecture for Class I graphs when t = 4 k + 2 for any integer k ≥ 1. -edge-connected (2 t +1)-regular Class I graphs without circular (2 t +2) /t -flows for any integer t ∈ { 6 , 8 , 10 } or t ≥ 12. Our result provides more general counterexamples to Steffen’s two conjectures for both even and odd t , and simultaneously generalizes the counterexamples of Jaeger’s Circular Flow Conjecture to regular Class I graphs.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2022-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Flow Index of Regular Class I Graphs\",\"authors\":\"Jiaao Li, Xueliang Li, Meiling Wang\",\"doi\":\"10.1137/21m1393169\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For integers k and d with k ≥ 2 d > 0, a circular k/d -flow of a graph G is an orientation together with a mapping from E ( G ) to {± d, ± ( d + 1) , . . . , ± ( k − d ) } such that, for each vertex of G , the sum of images on outgoing edges is equal to the sum of images on incoming edges. Related to the Four Color Problem, a classical result of Tutte shows that a cubic graph admits a circular 4 / 1-flow if and only if it is Class I (i.e., 3-edge-colorable). Tutte’s 3-flow conjecture implies that every 5-regular Class I graph admits a nowhere-zero 3-flow (equivalently, a circular 6 / 2-flow) as a special case. Steffen in 2015 conjectured that every (2 t + 1)-regular Class I graph admits a circular (2 t + 2) /t -flow. He also proposed a more general conjecture that every (2 t + 1)-odd-edge-connected (2 t + 1)-regular graph admits a circular (2 t + 2) /t -flow for any integer t ≥ 2, which includes the Circular Flow Conjecture of Jaeger(1981) stating that every 2 t -edge-connected graph admits a circular (2 t + 2) /t -flow for any even t ≥ 2. Jaeger’s conjecture was disproved in 2018 for all even t ≥ 6, and based on these results, Mattiolo and Steffen recently constructed counterexamples to Steffen’s conjecture for Class I graphs when t = 4 k + 2 for any integer k ≥ 1. -edge-connected (2 t +1)-regular Class I graphs without circular (2 t +2) /t -flows for any integer t ∈ { 6 , 8 , 10 } or t ≥ 12. Our result provides more general counterexamples to Steffen’s two conjectures for both even and odd t , and simultaneously generalizes the counterexamples of Jaeger’s Circular Flow Conjecture to regular Class I graphs.\",\"PeriodicalId\":21749,\"journal\":{\"name\":\"SIAM J. Discret. Math.\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-08-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"SIAM J. Discret. Math.\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1137/21m1393169\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM J. Discret. Math.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1137/21m1393169","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
For integers k and d with k ≥ 2 d > 0, a circular k/d -flow of a graph G is an orientation together with a mapping from E ( G ) to {± d, ± ( d + 1) , . . . , ± ( k − d ) } such that, for each vertex of G , the sum of images on outgoing edges is equal to the sum of images on incoming edges. Related to the Four Color Problem, a classical result of Tutte shows that a cubic graph admits a circular 4 / 1-flow if and only if it is Class I (i.e., 3-edge-colorable). Tutte’s 3-flow conjecture implies that every 5-regular Class I graph admits a nowhere-zero 3-flow (equivalently, a circular 6 / 2-flow) as a special case. Steffen in 2015 conjectured that every (2 t + 1)-regular Class I graph admits a circular (2 t + 2) /t -flow. He also proposed a more general conjecture that every (2 t + 1)-odd-edge-connected (2 t + 1)-regular graph admits a circular (2 t + 2) /t -flow for any integer t ≥ 2, which includes the Circular Flow Conjecture of Jaeger(1981) stating that every 2 t -edge-connected graph admits a circular (2 t + 2) /t -flow for any even t ≥ 2. Jaeger’s conjecture was disproved in 2018 for all even t ≥ 6, and based on these results, Mattiolo and Steffen recently constructed counterexamples to Steffen’s conjecture for Class I graphs when t = 4 k + 2 for any integer k ≥ 1. -edge-connected (2 t +1)-regular Class I graphs without circular (2 t +2) /t -flows for any integer t ∈ { 6 , 8 , 10 } or t ≥ 12. Our result provides more general counterexamples to Steffen’s two conjectures for both even and odd t , and simultaneously generalizes the counterexamples of Jaeger’s Circular Flow Conjecture to regular Class I graphs.