三角形连接有符号图上的整数流

Pub Date : 2024-02-08 DOI:10.1002/jgt.23076
Liangchen Li, Chong Li, Rong Luo, Cun-Quan Zhang
{"title":"三角形连接有符号图上的整数流","authors":"Liangchen Li,&nbsp;Chong Li,&nbsp;Rong Luo,&nbsp;Cun-Quan Zhang","doi":"10.1002/jgt.23076","DOIUrl":null,"url":null,"abstract":"<p>A triangle-path in a graph <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> is a sequence of distinct triangles <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>T</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>,</mo>\n \n <msub>\n <mi>T</mi>\n \n <mn>2</mn>\n </msub>\n \n <mo>,</mo>\n \n <mi>…</mi>\n \n <mo>,</mo>\n \n <msub>\n <mi>T</mi>\n \n <mi>m</mi>\n </msub>\n </mrow>\n <annotation> ${T}_{1},{T}_{2},\\ldots ,{T}_{m}$</annotation>\n </semantics></math> in <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> such that for any <span></span><math>\n <semantics>\n <mrow>\n <mi>i</mi>\n \n <mo>,</mo>\n \n <mi>j</mi>\n </mrow>\n <annotation> $i,j$</annotation>\n </semantics></math> with <span></span><math>\n <semantics>\n <mrow>\n <mn>1</mn>\n \n <mo>≤</mo>\n \n <mi>i</mi>\n \n <mo>&lt;</mo>\n \n <mi>j</mi>\n \n <mo>≤</mo>\n \n <mi>m</mi>\n </mrow>\n <annotation> $1\\le i\\lt j\\le m$</annotation>\n </semantics></math>, <span></span><math>\n <semantics>\n <mrow>\n <mo>∣</mo>\n \n <mi>E</mi>\n <mrow>\n <mo>(</mo>\n \n <msub>\n <mi>T</mi>\n \n <mi>i</mi>\n </msub>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∩</mo>\n \n <mi>E</mi>\n <mrow>\n <mo>(</mo>\n \n <msub>\n <mi>T</mi>\n <mrow>\n <mi>i</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </msub>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∣</mo>\n \n <mo>=</mo>\n \n <mn>1</mn>\n </mrow>\n <annotation> $| E({T}_{i})\\cap E({T}_{i+1})| =1$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n <mi>E</mi>\n <mrow>\n <mo>(</mo>\n \n <msub>\n <mi>T</mi>\n \n <mi>i</mi>\n </msub>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∩</mo>\n \n <mi>E</mi>\n <mrow>\n <mo>(</mo>\n \n <msub>\n <mi>T</mi>\n \n <mi>j</mi>\n </msub>\n \n <mo>)</mo>\n </mrow>\n \n <mo>=</mo>\n \n <mi>∅</mi>\n </mrow>\n <annotation> $E({T}_{i})\\cap E({T}_{j})=\\varnothing $</annotation>\n </semantics></math> if <span></span><math>\n <semantics>\n <mrow>\n <mi>j</mi>\n \n <mo>&gt;</mo>\n \n <mi>i</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n <annotation> $j\\gt i+1$</annotation>\n </semantics></math>. A connected graph <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> is triangularly connected if for any two nonparallel edges <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 <mi>e</mi>\n \n <mo>′</mo>\n </mrow>\n <annotation> $e^{\\prime} $</annotation>\n </semantics></math> there is a triangle-path <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>T</mi>\n \n <mn>1</mn>\n </msub>\n \n <msub>\n <mi>T</mi>\n \n <mn>2</mn>\n </msub>\n \n <mi>⋯</mi>\n \n <msub>\n <mi>T</mi>\n \n <mi>m</mi>\n </msub>\n </mrow>\n <annotation> ${T}_{1}{T}_{2}\\cdots {T}_{m}$</annotation>\n </semantics></math> such that <span></span><math>\n <semantics>\n <mrow>\n <mi>e</mi>\n \n <mo>∈</mo>\n \n <mi>E</mi>\n <mrow>\n <mo>(</mo>\n \n <msub>\n <mi>T</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $e\\in E({T}_{1})$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n <mi>e</mi>\n \n <mo>′</mo>\n \n <mo>∈</mo>\n \n <mi>E</mi>\n <mrow>\n <mo>(</mo>\n \n <msub>\n <mi>T</mi>\n \n <mi>m</mi>\n </msub>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $e^{\\prime} \\in E({T}_{m})$</annotation>\n </semantics></math>. For ordinary graphs, Fan et al. characterize all triangularly connected graphs that admit nowhere-zero 3-flows or 4-flows. Corollaries of this result include the integer flow of some families of ordinary graphs, such as locally connected graphs due to Lai and some types of products of graphs due to Imrich et al. In this paper, Fan's result for triangularly connected graphs is further extended to signed graphs. We proved that a flow-admissible triangularly connected signed graph admits a nowhere-zero 4-flow if and only if it is not the wheel <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>W</mi>\n \n <mn>5</mn>\n </msub>\n </mrow>\n <annotation> ${W}_{5}$</annotation>\n </semantics></math> associated with a specific signature. Moreover, this result is sharp since there are infinitely many unbalanced triangularly connected signed graphs admitting a nowhere-zero 4-flow but no 3-flow.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Integer flows on triangularly connected signed graphs\",\"authors\":\"Liangchen Li,&nbsp;Chong Li,&nbsp;Rong Luo,&nbsp;Cun-Quan Zhang\",\"doi\":\"10.1002/jgt.23076\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A triangle-path in a graph <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> is a sequence of distinct triangles <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>T</mi>\\n \\n <mn>1</mn>\\n </msub>\\n \\n <mo>,</mo>\\n \\n <msub>\\n <mi>T</mi>\\n \\n <mn>2</mn>\\n </msub>\\n \\n <mo>,</mo>\\n \\n <mi>…</mi>\\n \\n <mo>,</mo>\\n \\n <msub>\\n <mi>T</mi>\\n \\n <mi>m</mi>\\n </msub>\\n </mrow>\\n <annotation> ${T}_{1},{T}_{2},\\\\ldots ,{T}_{m}$</annotation>\\n </semantics></math> in <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> such that for any <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>i</mi>\\n \\n <mo>,</mo>\\n \\n <mi>j</mi>\\n </mrow>\\n <annotation> $i,j$</annotation>\\n </semantics></math> with <span></span><math>\\n <semantics>\\n <mrow>\\n <mn>1</mn>\\n \\n <mo>≤</mo>\\n \\n <mi>i</mi>\\n \\n <mo>&lt;</mo>\\n \\n <mi>j</mi>\\n \\n <mo>≤</mo>\\n \\n <mi>m</mi>\\n </mrow>\\n <annotation> $1\\\\le i\\\\lt j\\\\le m$</annotation>\\n </semantics></math>, <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>∣</mo>\\n \\n <mi>E</mi>\\n <mrow>\\n <mo>(</mo>\\n \\n <msub>\\n <mi>T</mi>\\n \\n <mi>i</mi>\\n </msub>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>∩</mo>\\n \\n <mi>E</mi>\\n <mrow>\\n <mo>(</mo>\\n \\n <msub>\\n <mi>T</mi>\\n <mrow>\\n <mi>i</mi>\\n \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n </msub>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>∣</mo>\\n \\n <mo>=</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n <annotation> $| E({T}_{i})\\\\cap E({T}_{i+1})| =1$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>E</mi>\\n <mrow>\\n <mo>(</mo>\\n \\n <msub>\\n <mi>T</mi>\\n \\n <mi>i</mi>\\n </msub>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>∩</mo>\\n \\n <mi>E</mi>\\n <mrow>\\n <mo>(</mo>\\n \\n <msub>\\n <mi>T</mi>\\n \\n <mi>j</mi>\\n </msub>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>=</mo>\\n \\n <mi>∅</mi>\\n </mrow>\\n <annotation> $E({T}_{i})\\\\cap E({T}_{j})=\\\\varnothing $</annotation>\\n </semantics></math> if <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>j</mi>\\n \\n <mo>&gt;</mo>\\n \\n <mi>i</mi>\\n \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n <annotation> $j\\\\gt i+1$</annotation>\\n </semantics></math>. A connected graph <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> is triangularly connected if for any two nonparallel edges <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 <mi>e</mi>\\n \\n <mo>′</mo>\\n </mrow>\\n <annotation> $e^{\\\\prime} $</annotation>\\n </semantics></math> there is a triangle-path <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>T</mi>\\n \\n <mn>1</mn>\\n </msub>\\n \\n <msub>\\n <mi>T</mi>\\n \\n <mn>2</mn>\\n </msub>\\n \\n <mi>⋯</mi>\\n \\n <msub>\\n <mi>T</mi>\\n \\n <mi>m</mi>\\n </msub>\\n </mrow>\\n <annotation> ${T}_{1}{T}_{2}\\\\cdots {T}_{m}$</annotation>\\n </semantics></math> such that <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>e</mi>\\n \\n <mo>∈</mo>\\n \\n <mi>E</mi>\\n <mrow>\\n <mo>(</mo>\\n \\n <msub>\\n <mi>T</mi>\\n \\n <mn>1</mn>\\n </msub>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $e\\\\in E({T}_{1})$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>e</mi>\\n \\n <mo>′</mo>\\n \\n <mo>∈</mo>\\n \\n <mi>E</mi>\\n <mrow>\\n <mo>(</mo>\\n \\n <msub>\\n <mi>T</mi>\\n \\n <mi>m</mi>\\n </msub>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $e^{\\\\prime} \\\\in E({T}_{m})$</annotation>\\n </semantics></math>. For ordinary graphs, Fan et al. characterize all triangularly connected graphs that admit nowhere-zero 3-flows or 4-flows. Corollaries of this result include the integer flow of some families of ordinary graphs, such as locally connected graphs due to Lai and some types of products of graphs due to Imrich et al. In this paper, Fan's result for triangularly connected graphs is further extended to signed graphs. We proved that a flow-admissible triangularly connected signed graph admits a nowhere-zero 4-flow if and only if it is not the wheel <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>W</mi>\\n \\n <mn>5</mn>\\n </msub>\\n </mrow>\\n <annotation> ${W}_{5}$</annotation>\\n </semantics></math> associated with a specific signature. Moreover, this result is sharp since there are infinitely many unbalanced triangularly connected signed graphs admitting a nowhere-zero 4-flow but no 3-flow.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-02-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23076\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23076","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

摘要

图 G$G$ 中的三角形路径是一连串不同的三角形 T1,T2,......。,Tm${T}_{1},{T}_{2},\ldots,{T}_{m}$在 G$G$中,对于任意 i,j$i,j$,1≤i<;j≤m$1\le i\lt j\le m$, ∣E(Ti)∩E(Ti+1)∣=1$| E({T}_{i})\cap E({T}_{i+1})| =1$ 和 E(Ti)∩E(Tj)=∅$E({T}_{i})\cap E({T}_{j})=\varnothing $ if j>i+1$j\gt i+1$.如果对于任意两条不平行的边 e$e$ 和 e′$e^{prime} $ 有一条三角形-路径 T1T2⋯Tm${T}_{1}{T}_{2}\cdots {T}_{m}$ ,使得 e∈E(T1)$e\in E({T}_{1})$ 和 e′∈E(Tm)$e^{prime}\in E({T}_{m})$.对于普通图,Fan 等人描述了所有允许无处为零的 3 流或 4 流的三角形连接图。这一结果的推论包括一些普通图族的整数流,如 Lai 提出的局部连通图和 Imrich 等人提出的某些类型的图积。我们证明,当且仅当一个流动可容许的三角形连接有符号图不是与特定签名相关的轮 W5${W}_{5}$时,它才容许一个无处为零的 4 流。此外,这个结果是尖锐的,因为有无限多的不平衡三角形连接有符号图允许无处为零的 4 流,但不允许 3 流。
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Integer flows on triangularly connected signed graphs

A triangle-path in a graph G $G$ is a sequence of distinct triangles T 1 , T 2 , , T m ${T}_{1},{T}_{2},\ldots ,{T}_{m}$ in G $G$ such that for any i , j $i,j$ with 1 i < j m $1\le i\lt j\le m$ , E ( T i ) E ( T i + 1 ) = 1 $| E({T}_{i})\cap E({T}_{i+1})| =1$ and E ( T i ) E ( T j ) = $E({T}_{i})\cap E({T}_{j})=\varnothing $ if j > i + 1 $j\gt i+1$ . A connected graph G $G$ is triangularly connected if for any two nonparallel edges e $e$ and e $e^{\prime} $ there is a triangle-path T 1 T 2 T m ${T}_{1}{T}_{2}\cdots {T}_{m}$ such that e E ( T 1 ) $e\in E({T}_{1})$ and e E ( T m ) $e^{\prime} \in E({T}_{m})$ . For ordinary graphs, Fan et al. characterize all triangularly connected graphs that admit nowhere-zero 3-flows or 4-flows. Corollaries of this result include the integer flow of some families of ordinary graphs, such as locally connected graphs due to Lai and some types of products of graphs due to Imrich et al. In this paper, Fan's result for triangularly connected graphs is further extended to signed graphs. We proved that a flow-admissible triangularly connected signed graph admits a nowhere-zero 4-flow if and only if it is not the wheel W 5 ${W}_{5}$ associated with a specific signature. Moreover, this result is sharp since there are infinitely many unbalanced triangularly connected signed graphs admitting a nowhere-zero 4-flow but no 3-flow.

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