正则I类图的流量指数

Jiaao Li, Xueliang Li, Meiling Wang
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引用次数: 0

摘要

对于k≥2 d > 0的整数k和d,图G的圆形k/d流是一个方向以及从E (G)到{±d,±(d + 1),…的映射。,±(k−d)},使得对于G的每个顶点,出边的图像之和等于进边的图像之和。与四色问题相关,Tutte的一个经典结果表明,当且仅当三次图是I类(即3边可着色)时,它允许圆4 / 1流。Tutte的3流猜想意味着,每一个5正则I类图都承认一个无零的3流(相当于一个圆形的6 / 2流)作为特殊情况。Steffen(2015)推测,每一个(2t + 1)-正则I类图都存在一个循环(2t + 2) /t -流。他还提出了一个更一般的猜想,即对于任何整数t≥2,每个(2t + 1)-奇边连通(2t + 1)-正则图都存在一个圆形(2t + 2) /t -流,其中包括Jaeger(1981)的圆形流猜想,即对于任何偶数t≥2,每个2t边连通图都存在一个圆形(2t + 2) /t -流。Jaeger的猜想在2018年被证明是错误的,对于所有偶数t≥6,基于这些结果,Mattiolo和Steffen最近为I类图构建了Steffen猜想的反例,当t = 4k + 2时,对于任何整数k≥1。-边连通(2t +1)-不含圆形(2t +2) /t的正则I类图-任意整数t∈{6,8,10}或t≥12的流。我们的结果为Steffen的偶数和奇数t猜想提供了更一般的反例,同时将Jaeger的循环流猜想的反例推广到正则I类图。
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The Flow Index of Regular Class I Graphs
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.
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