Circular flows in mono-directed signed graphs

IF 0.9 3区 数学 Q2 MATHEMATICS
Jiaao Li, Reza Naserasr, Zhouningxin Wang, Xuding Zhu
{"title":"Circular flows in mono-directed signed graphs","authors":"Jiaao Li,&nbsp;Reza Naserasr,&nbsp;Zhouningxin Wang,&nbsp;Xuding Zhu","doi":"10.1002/jgt.23092","DOIUrl":null,"url":null,"abstract":"<p>In this paper, the concept of circular <span></span><math>\n <semantics>\n <mrow>\n <mi>r</mi>\n </mrow>\n <annotation> $r$</annotation>\n </semantics></math>-flow in a mono-directed signed graph <span></span><math>\n <semantics>\n <mrow>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>G</mi>\n <mo>,</mo>\n <mi>σ</mi>\n </mrow>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $(G,\\sigma )$</annotation>\n </semantics></math> is introduced. That is a pair <span></span><math>\n <semantics>\n <mrow>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>D</mi>\n <mo>,</mo>\n <mi>f</mi>\n </mrow>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $(D,f)$</annotation>\n </semantics></math>, where <span></span><math>\n <semantics>\n <mrow>\n <mi>D</mi>\n </mrow>\n <annotation> $D$</annotation>\n </semantics></math> is an orientation on <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n <mi>f</mi>\n <mo>:</mo>\n <mi>E</mi>\n <mrow>\n <mo>(</mo>\n <mi>G</mi>\n <mo>)</mo>\n </mrow>\n <mo>→</mo>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mo>−</mo>\n <mi>r</mi>\n <mo>,</mo>\n <mi>r</mi>\n </mrow>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $f:E(G)\\to (-r,r)$</annotation>\n </semantics></math> satisfies that <span></span><math>\n <semantics>\n <mrow>\n <mo>∣</mo>\n <mi>f</mi>\n <mrow>\n <mo>(</mo>\n <mi>e</mi>\n <mo>)</mo>\n </mrow>\n <mo>∣</mo>\n <mo>∈</mo>\n <mrow>\n <mo>[</mo>\n <mrow>\n <mn>1</mn>\n <mo>,</mo>\n <mi>r</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n <mo>]</mo>\n </mrow>\n </mrow>\n <annotation> $| f(e)| \\in [1,r-1]$</annotation>\n </semantics></math> for each positive edge <span></span><math>\n <semantics>\n <mrow>\n <mi>e</mi>\n </mrow>\n <annotation> $e$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n <mo>∣</mo>\n <mi>f</mi>\n <mrow>\n <mo>(</mo>\n <mi>e</mi>\n <mo>)</mo>\n </mrow>\n <mo>∣</mo>\n <mo>∈</mo>\n <mrow>\n <mo>[</mo>\n <mrow>\n <mn>0</mn>\n <mo>,</mo>\n <mfrac>\n <mi>r</mi>\n <mn>2</mn>\n </mfrac>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n <mo>]</mo>\n </mrow>\n <mo>∪</mo>\n <mrow>\n <mo>[</mo>\n <mrow>\n <mfrac>\n <mi>r</mi>\n <mn>2</mn>\n </mfrac>\n <mo>+</mo>\n <mn>1</mn>\n <mo>,</mo>\n <mi>r</mi>\n </mrow>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $| f(e)| \\in [0,\\frac{r}{2}-1]\\cup [\\frac{r}{2}+1,r)$</annotation>\n </semantics></math> for each negative edge <span></span><math>\n <semantics>\n <mrow>\n <mi>e</mi>\n </mrow>\n <annotation> $e$</annotation>\n </semantics></math>, and the total in-flow equals the total out-flow at each vertex. This is the dual notion of circular colorings of signed graphs and is distinct from the concept of circular flows in bi-directed graphs associated with signed graphs studied in the literature. We first explore the connection between circular <span></span><math>\n <semantics>\n <mrow>\n <mfrac>\n <mrow>\n <mn>2</mn>\n <mi>k</mi>\n </mrow>\n <mrow>\n <mi>k</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </mfrac>\n </mrow>\n <annotation> $\\frac{2k}{k-1}$</annotation>\n </semantics></math>-flows and modulo <span></span><math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-orientations in signed graphs. Then we focus on the upper bounds for the circular flow indices of signed graphs in terms of the edge-connectivity, where the circular flow index of a signed graph is the minimum value <span></span><math>\n <semantics>\n <mrow>\n <mi>r</mi>\n </mrow>\n <annotation> $r$</annotation>\n </semantics></math> such that it admits a circular <span></span><math>\n <semantics>\n <mrow>\n <mi>r</mi>\n </mrow>\n <annotation> $r$</annotation>\n </semantics></math>-flow. We prove that every 3-edge-connected signed graph admits a circular 6-flow and every 4-edge-connected signed graph admits a circular 4-flow. More generally, for <span></span><math>\n <semantics>\n <mrow>\n <mi>k</mi>\n <mo>≥</mo>\n <mn>2</mn>\n </mrow>\n <annotation> $k\\ge 2$</annotation>\n </semantics></math>, we show that every <span></span><math>\n <semantics>\n <mrow>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mn>3</mn>\n <mi>k</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $(3k-1)$</annotation>\n </semantics></math>-edge-connected signed graph admits a circular <span></span><math>\n <semantics>\n <mrow>\n <mfrac>\n <mrow>\n <mn>2</mn>\n <mi>k</mi>\n </mrow>\n <mrow>\n <mi>k</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </mfrac>\n </mrow>\n <annotation> $\\frac{2k}{k-1}$</annotation>\n </semantics></math>-flow, every <span></span><math>\n <semantics>\n <mrow>\n <mn>3</mn>\n <mi>k</mi>\n </mrow>\n <annotation> $3k$</annotation>\n </semantics></math>-edge-connected signed graph has a circular <span></span><math>\n <semantics>\n <mrow>\n <mi>r</mi>\n </mrow>\n <annotation> $r$</annotation>\n </semantics></math>-flow with <span></span><math>\n <semantics>\n <mrow>\n <mi>r</mi>\n <mo>&lt;</mo>\n <mfrac>\n <mrow>\n <mn>2</mn>\n <mi>k</mi>\n </mrow>\n <mrow>\n <mi>k</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </mfrac>\n </mrow>\n <annotation> $r\\lt \\frac{2k}{k-1}$</annotation>\n </semantics></math>, and every <span></span><math>\n <semantics>\n <mrow>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mn>3</mn>\n <mi>k</mi>\n <mo>+</mo>\n <mn>1</mn>\n </mrow>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $(3k+1)$</annotation>\n </semantics></math>-edge-connected signed graph admits a circular <span></span><math>\n <semantics>\n <mrow>\n <mfrac>\n <mrow>\n <mn>4</mn>\n <mi>k</mi>\n <mo>+</mo>\n <mn>2</mn>\n </mrow>\n <mrow>\n <mn>2</mn>\n <mi>k</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </mfrac>\n </mrow>\n <annotation> $\\frac{4k+2}{2k-1}$</annotation>\n </semantics></math>-flow. Moreover, the <span></span><math>\n <semantics>\n <mrow>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mn>6</mn>\n <mi>k</mi>\n <mo>−</mo>\n <mn>2</mn>\n </mrow>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $(6k-2)$</annotation>\n </semantics></math>-edge-connectivity condition is shown to be sufficient for a signed Eulerian graph to admit a circular <span></span><math>\n <semantics>\n <mrow>\n <mfrac>\n <mrow>\n <mn>4</mn>\n <mi>k</mi>\n </mrow>\n <mrow>\n <mn>2</mn>\n <mi>k</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </mfrac>\n </mrow>\n <annotation> $\\frac{4k}{2k-1}$</annotation>\n </semantics></math>-flow, and applying this result to planar graphs, we conclude that every signed bipartite planar graph of negative girth at least <span></span><math>\n <semantics>\n <mrow>\n <mn>6</mn>\n <mi>k</mi>\n <mo>−</mo>\n <mn>2</mn>\n </mrow>\n <annotation> $6k-2$</annotation>\n </semantics></math> admits a homomorphism to the negative even cycles <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>C</mi>\n <mrow>\n <mo>−</mo>\n <mn>2</mn>\n <mi>k</mi>\n </mrow>\n </msub>\n </mrow>\n <annotation> ${C}_{-2k}$</annotation>\n </semantics></math>.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"106 3","pages":"686-710"},"PeriodicalIF":0.9000,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23092","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

In this paper, the concept of circular r $r$ -flow in a mono-directed signed graph ( G , σ ) $(G,\sigma )$ is introduced. That is a pair ( D , f ) $(D,f)$ , where D $D$ is an orientation on G $G$ and f : E ( G ) ( r , r ) $f:E(G)\to (-r,r)$ satisfies that f ( e ) [ 1 , r 1 ] $| f(e)| \in [1,r-1]$ for each positive edge e $e$ and f ( e ) [ 0 , r 2 1 ] [ r 2 + 1 , r ) $| f(e)| \in [0,\frac{r}{2}-1]\cup [\frac{r}{2}+1,r)$ for each negative edge e $e$ , and the total in-flow equals the total out-flow at each vertex. This is the dual notion of circular colorings of signed graphs and is distinct from the concept of circular flows in bi-directed graphs associated with signed graphs studied in the literature. We first explore the connection between circular 2 k k 1 $\frac{2k}{k-1}$ -flows and modulo k $k$ -orientations in signed graphs. Then we focus on the upper bounds for the circular flow indices of signed graphs in terms of the edge-connectivity, where the circular flow index of a signed graph is the minimum value r $r$ such that it admits a circular r $r$ -flow. We prove that every 3-edge-connected signed graph admits a circular 6-flow and every 4-edge-connected signed graph admits a circular 4-flow. More generally, for k 2 $k\ge 2$ , we show that every ( 3 k 1 ) $(3k-1)$ -edge-connected signed graph admits a circular 2 k k 1 $\frac{2k}{k-1}$ -flow, every 3 k $3k$ -edge-connected signed graph has a circular r $r$ -flow with r < 2 k k 1 $r\lt \frac{2k}{k-1}$ , and every ( 3 k + 1 ) $(3k+1)$ -edge-connected signed graph admits a circular 4 k + 2 2 k 1 $\frac{4k+2}{2k-1}$ -flow. Moreover, the ( 6 k 2 ) $(6k-2)$ -edge-connectivity condition is shown to be sufficient for a signed Eulerian graph to admit a circular 4 k 2 k 1 $\frac{4k}{2k-1}$ -flow, and applying this result to planar graphs, we conclude that every signed bipartite planar graph of negative girth at least 6 k 2 $6k-2$ admits a homomorphism to the negative even cycles C 2 k ${C}_{-2k}$ .

单向有符号图中的循环流动
本文引入了单向有符号图(G,σ)$(G,\sigma )$中循环 r$r$ 流的概念。即一对 (D,f)$(D,f)$,其中 D$D$ 是 G$G$ 上的方向,f:E(G)→(-r,r)$f:E(G)\to(-r,r)$满足∣f(e)∣∈[1,r-1]$| f(e)| \in [1,r-1]$ for each positive edge e$e$ and ∣f(e)∣∈[0,r2-1]∪[r2+1、r)$|f(e)|\in[0,\frac{r}{2}-1]\cup [\frac{r}{2}+1,r)$对于每条负边 e$e$,总流入等于每个顶点的总流出。这就是有符号图的循环着色的对偶概念,有别于文献中研究的与有符号图相关的双向图中的循环流概念。我们首先探讨了有符号图中循环 2kk-1$\frac{2k}{k-1}$ 流与 modulo k$k$-orientation 之间的联系。有符号图的循环流指数是允许循环 r$r$ 流的最小值 r$r$。我们证明了每一个 3 边连接的有符号图都允许循环 6 流,每一个 4 边连接的有符号图都允许循环 4 流。更一般地说,对于 k≥2$k\ge 2$,我们证明每一个 (3k-1)$(3k-1)$ 边连接的有符号图都有一个环形 2kk-1$frac{2k}{k-1}$ 流,每一个 3k$3k$ 边连接的有符号图都有一个环形 r$r$ 流,r<;2kk-1$r\lt \frac{2k}{k-1}$,而每个 (3k+1)$(3k+1)$ 边连接的有符号图都有一个循环的 4k+22k-1$frac{4k+2}{2k-1}$ 流。此外,(6k-2)$(6k-2)$-边连接条件被证明足以让有符号欧拉图接纳循环 4k2k-1$/frac{4k}{2k-1}$-流,将这一结果应用于平面图,我们得出结论:每个负周长至少为 6k-2$6k-2$ 的有符号双方平面图都接纳与负偶数循环 C-2k${C}_{-2k}$ 的同构。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Journal of Graph Theory
Journal of Graph Theory 数学-数学
CiteScore
1.60
自引率
22.20%
发文量
130
审稿时长
6-12 weeks
期刊介绍: The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .
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