We study a counting version of Cycle Double Cover Conjecture. We discuss why it is more interesting to count circuits (i.e., graphs isomorphic to for some ) instead of cycles (graphs with all degrees even). We give an almost-exponential lower bound for graphs with a surface embedding of representativity at least 4. We also prove an exponential lower bound for planar graphs. We conjecture that any bridgeless cubic graph has at least circuit double covers and we show an infinite class of graphs for which this bound is tight.
{"title":"Counting circuit double covers","authors":"Radek Hušek, Robert Šámal","doi":"10.1002/jgt.23187","DOIUrl":"https://doi.org/10.1002/jgt.23187","url":null,"abstract":"<p>We study a counting version of Cycle Double Cover Conjecture. We discuss why it is more interesting to count circuits (i.e., graphs isomorphic to <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>C</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${C}_{k}$</annotation>\u0000 </semantics></math> for some <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>) instead of cycles (graphs with all degrees even). We give an almost-exponential lower bound for graphs with a surface embedding of representativity at least 4. We also prove an exponential lower bound for planar graphs. We conjecture that any bridgeless cubic graph has at least <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mn>2</mn>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>/</mo>\u0000 \u0000 <mn>2</mn>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${2}^{n/2-1}$</annotation>\u0000 </semantics></math> circuit double covers and we show an infinite class of graphs for which this bound is tight.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 2","pages":"374-395"},"PeriodicalIF":0.9,"publicationDate":"2024-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23187","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142868031","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note it is proven that there exist infinitely many positive integers and such that the Cartesian product of a directed cycle of length and a directed cycle of length is non-Hamiltonian. In particular, the Cartesian product of an 880-dicycle and a 4368-dicycle is non-Hamiltonian. We also prove that there is no such graph on fewer than vertices, which is rather astonishing.
本论文证明了存在无穷多个正整数 a $a$ 和 b $b$ ，使得长度为 2 a $2a$ 的有向循环和长度为 2 b $2b$ 的有向循环的笛卡儿积为非哈密尔顿积。特别是，880-圆周和 4368-圆周的笛卡儿积为非哈密尔顿积。我们还证明了在少于 880 ⋅ 4368 = 3 , 843 , 840 $880cdot 4368=3,843,840$ 的顶点上不存在这样的图，这是相当惊人的。
{"title":"Non-Hamiltonian Cartesian products of two even dicycles","authors":"Kenta Noguchi, Carol T. Zamfirescu","doi":"10.1002/jgt.23185","DOIUrl":"https://doi.org/10.1002/jgt.23185","url":null,"abstract":"<p>In this note it is proven that there exist infinitely many positive integers <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 </mrow>\u0000 <annotation> $a$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>b</mi>\u0000 </mrow>\u0000 <annotation> $b$</annotation>\u0000 </semantics></math> such that the Cartesian product of a directed cycle of length <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>a</mi>\u0000 </mrow>\u0000 <annotation> $2a$</annotation>\u0000 </semantics></math> and a directed cycle of length <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>b</mi>\u0000 </mrow>\u0000 <annotation> $2b$</annotation>\u0000 </semantics></math> is non-Hamiltonian. In particular, the Cartesian product of an 880-dicycle and a 4368-dicycle is non-Hamiltonian. We also prove that there is no such graph on fewer than <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>880</mn>\u0000 \u0000 <mo>⋅</mo>\u0000 \u0000 <mn>4368</mn>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>3</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>843</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>840</mn>\u0000 </mrow>\u0000 <annotation> $880cdot 4368=3,843,840$</annotation>\u0000 </semantics></math> vertices, which is rather astonishing.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 2","pages":"368-373"},"PeriodicalIF":0.9,"publicationDate":"2024-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142869244","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<p>The celebrated Nash-Williams and Tutte's Theorem states that a graph <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math> contains <span></span><math> <semantics> <mrow> <mrow> <mi>k</mi> </mrow> </mrow> <annotation> $k$</annotation> </semantics></math> edge-disjoint spanning trees if and only if <span></span><math> <semantics> <mrow> <mrow> <msub> <mi>ν</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>≥</mo> <mi>k</mi> </mrow> </mrow> <annotation> ${nu }_{f}(G)ge k$</annotation> </semantics></math>, where </p><p>In this paper, we prove that, for integers <span></span><math> <semantics> <mrow> <mrow> <mi>k</mi> <mo>≥</mo> <mn>0</mn> </mrow> </mrow> <annotation> $kge 0$</annotation> </semantics></math>, <span></span><math> <semantics> <mrow> <mrow> <mi>d</mi> <mo>≥</mo> <mn>1</mn> </mrow> </mrow> <annotation> $dge 1$</annotation> </semantics></math>, if <span></span><math> <semantics> <mrow> <mrow> <msub> <mi>ν</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>></mo> <mi>k</mi> <mo>+</mo> <mfrac> <mrow>
著名的 Nash-Williams 和 Tutte 定理指出，当且仅当 ν f ( G ) ≥ k ${{nu }_{f}(G)ge k$ 时，图 G $G$ 包含 k $k$ 边互不相交的生成树，在本文中，我们证明，对于整数 k ≥ 0 $kge 0$ , d ≥ 1 $dge 1$ , 如果 ν f ( G ) > k + d - 1 d ${{nu }_{f}(G)gt k+frac{d-1}{d}$ ，则 G $G$ 包含 k $k$ 边互不相交的生成树和另一个森林 F $F$ ，其 ∣ E ( F ) ∣ > d - 1 d ( ∣ V ( G ) ∣ - 1 ) $| E(F)| gt frac{d-1}{d}(|V(G)|-1)$，如果 F $F$ 不是生成树，那么 F $F$ 有一个至少有 d $d$ 边的部分。
{"title":"An extension of Nash-Williams and Tutte's Theorem","authors":"Xuqian Fang, Daqing Yang","doi":"10.1002/jgt.23189","DOIUrl":"https://doi.org/10.1002/jgt.23189","url":null,"abstract":"<p>The celebrated Nash-Williams and Tutte's Theorem states that a graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> contains <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math> edge-disjoint spanning trees if and only if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>ν</mi>\u0000 \u0000 <mi>f</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${nu }_{f}(G)ge k$</annotation>\u0000 </semantics></math>, where\u0000\u0000 </p><p>In this paper, we prove that, for integers <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>0</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $kge 0$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>d</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $dge 1$</annotation>\u0000 </semantics></math>, if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>ν</mi>\u0000 \u0000 <mi>f</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>></mo>\u0000 \u0000 <mi>k</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mfrac>\u0000 <mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 2","pages":"361-367"},"PeriodicalIF":0.9,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142869216","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In recent years, discrete notions of curvature have been defined and exploited to understand various geometric properties of graphs; especially regarding heat flow, and spectral properties. In this paper, we study various combinatorial properties implied by satisfying the Bakry–Émery curvature dimension inequality . In particular we derive a local discrepancy inequality, similar in spirit to the expander mixing lemma from spectral graph theory, which certifies a type of “local pseudo-randomness” of the edge set of the graph, for graphs satisfying a curvature lower bound. In addition, several other consequences are derived regarding graph connectivity and cycle statistics of the graph.
{"title":"Graph curvature and local discrepancy","authors":"Paul Horn, Adam Purcilly, Alex Stevens","doi":"10.1002/jgt.23176","DOIUrl":"https://doi.org/10.1002/jgt.23176","url":null,"abstract":"<p>In recent years, discrete notions of curvature have been defined and exploited to understand various geometric properties of graphs; especially regarding heat flow, and spectral properties. In this paper, we study various combinatorial properties implied by satisfying the Bakry–Émery curvature dimension inequality <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>C</mi>\u0000 \u0000 <mi>D</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>∞</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>K</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $CD(infty ,K)$</annotation>\u0000 </semantics></math>. In particular we derive a local discrepancy inequality, similar in spirit to the expander mixing lemma from spectral graph theory, which certifies a type of “local pseudo-randomness” of the edge set of the graph, for graphs satisfying a curvature lower bound. In addition, several other consequences are derived regarding graph connectivity and cycle statistics of the graph.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 2","pages":"337-360"},"PeriodicalIF":0.9,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142869195","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<p>A generalization of the famous Caccetta–Häggkvist conjecture, suggested by Aharoni, is that any family <span></span><math> <semantics> <mrow> <mi>F</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>F</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>…</mi> <mo>,</mo> <msub> <mi>F</mi> <mi>n</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> ${rm{ {mathcal F} }}=({F}_{1},ldots ,{F}_{n})$</annotation> </semantics></math> of sets of edges in <span></span><math> <semantics> <mrow> <msub> <mi>K</mi> <mi>n</mi> </msub> </mrow> <annotation> ${K}_{n}$</annotation> </semantics></math>, each of size <span></span><math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math>, has a rainbow cycle of length at most <span></span><math> <semantics> <mrow> <mo>⌈</mo> <mfrac> <mi>n</mi> <mi>k</mi> </mfrac> <mo>⌉</mo> </mrow> <annotation> $lceil frac{n}{k}rceil $</annotation> </semantics></math>. In works by the author with Aharoni and by the author with Aharoni, Berger, Chudnovsky, and Zerbib, it was shown that asymptotically this can be improved to <span></span><math> <semantics> <mrow> <mi>O</mi> <mrow> <mo>(</mo> <mrow> <mi>log</mi> <mi>n</mi> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> $O(mathrm{log}n)$</annotation> </se
由阿哈罗尼提出的著名的卡塞塔-哈格克维斯特猜想的一个概括是：任何族 F = ( F 1 , ... , F n ) ${rm{ {mathcal F} }}=({F}_{1},ldots ,{F}_{n})$ K n ${K}_{n} 中的边集，每个边集的大小为 k $k}}=({F}_{1},ldots ,{F}_{n})$ K n ${K}_{n}$ 中的边集，每个边集的大小为 k $k$，最多有⌈ n k ⌉ $lceil frac{n}{k}rceil $ 长度的彩虹循环。作者与 Aharoni 以及作者与 Aharoni、Berger、Chudnovsky 和 Zerbib 的研究表明，如果所有集合都是大小为 2 的匹配集，或者所有集合都是三角形，那么从渐近的角度来看，这个结果可以改进为 O ( log n ) $O(mathrm{log}n)$。我们将证明，在混合情况下，即每个 F i ${F}_{i}$ 要么是大小为 2 的匹配集要么是三角形时，情况也是如此。我们还研究了每个 F i ${F}_{i}$ 都是大小为 2 的匹配或单边，或者每个 F i ${F}_{i}$ 都是三角形或单边的情况，在每种情况下，我们都确定了类型之间的临界比例，超过这个比例，彩虹周长就会从线性变为对数。
{"title":"Short rainbow cycles for families of matchings and triangles","authors":"He Guo","doi":"10.1002/jgt.23183","DOIUrl":"https://doi.org/10.1002/jgt.23183","url":null,"abstract":"<p>A generalization of the famous Caccetta–Häggkvist conjecture, suggested by Aharoni, is that any family <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 \u0000 <mo>=</mo>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <msub>\u0000 <mi>F</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>…</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>F</mi>\u0000 \u0000 <mi>n</mi>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{ {mathcal F} }}=({F}_{1},ldots ,{F}_{n})$</annotation>\u0000 </semantics></math> of sets of edges in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mi>n</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${K}_{n}$</annotation>\u0000 </semantics></math>, each of size <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>, has a rainbow cycle of length at most <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>⌈</mo>\u0000 \u0000 <mfrac>\u0000 <mi>n</mi>\u0000 \u0000 <mi>k</mi>\u0000 </mfrac>\u0000 \u0000 <mo>⌉</mo>\u0000 </mrow>\u0000 <annotation> $lceil frac{n}{k}rceil $</annotation>\u0000 </semantics></math>. In works by the author with Aharoni and by the author with Aharoni, Berger, Chudnovsky, and Zerbib, it was shown that asymptotically this can be improved to <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>O</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>log</mi>\u0000 \u0000 <mi>n</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $O(mathrm{log}n)$</annotation>\u0000 </se","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 2","pages":"325-336"},"PeriodicalIF":0.9,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23183","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142868944","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<p>In this short note, we consider a graph process recently introduced by Frieze, Krivelevich and Michaeli. In their model, the edges of the complete graph <span></span><math> <semantics> <mrow> <msub> <mi>K</mi> <mi>n</mi> </msub> </mrow> <annotation> ${K}_{n}$</annotation> </semantics></math> are ordered uniformly at random and are then revealed consecutively to a player called Builder. At every round, Builder must decide if they accept the edge proposed at this round or not. We prove that, for every <span></span><math> <semantics> <mrow> <mi>d</mi> <mo>≥</mo> <mn>2</mn> </mrow> <annotation> $dge 2$</annotation> </semantics></math>, Builder can construct a spanning <span></span><math> <semantics> <mrow> <mi>d</mi> </mrow> <annotation> $d$</annotation> </semantics></math>-connected graph after <span></span><math> <semantics> <mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mi>o</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mi>n</mi> <mi>log</mi> <mo> </mo> <mi>n</mi> <mo>/</mo> <mn>2</mn> </mrow> <annotation> $(1+o(1))nmathrm{log}unicode{x0200A}n/2$</annotation> </semantics></math> rounds by accepting <span></span><math> <semantics> <mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mi>o</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>n</mi>
在这篇短文中，我们考虑Frieze、Krivelevich和Michaeli最近引入的一个图形过程。在他们的模型中，完整图K n ${K}_{n}$的边是随机排列的，然后连续地显示给一个叫做Builder的玩家。在每一轮，建设者必须决定他们是否接受在这一轮提出的优势。我们证明，对于每一个d≥2 $dge 2$，Builder可以在(1 + 0(1)之后构造一个生成d $d$连通图)) n log n / 2 $(1+o(1))nmathrm{log}unicode{x0200A}n/2$通过接受1 + 0 (1)) d n / 2$(1+o(1))dn/2$当n→∞时概率收敛到1的边$nto infty $。这就解决了弗里兹、克里维列维奇和米切利的一个猜想。
{"title":"d\u0000 \u0000 $d$\u0000 -connectivity of the random graph with restricted budget","authors":"Lyuben Lichev","doi":"10.1002/jgt.23180","DOIUrl":"https://doi.org/10.1002/jgt.23180","url":null,"abstract":"<p>In this short note, we consider a graph process recently introduced by Frieze, Krivelevich and Michaeli. In their model, the edges of the complete graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mi>n</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${K}_{n}$</annotation>\u0000 </semantics></math> are ordered uniformly at random and are then revealed consecutively to a player called Builder. At every round, Builder must decide if they accept the edge proposed at this round or not. We prove that, for every <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation> $dge 2$</annotation>\u0000 </semantics></math>, Builder can construct a spanning <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation> $d$</annotation>\u0000 </semantics></math>-connected graph after <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mi>o</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mi>log</mi>\u0000 \u0000 <mo> </mo>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>/</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation> $(1+o(1))nmathrm{log}unicode{x0200A}n/2$</annotation>\u0000 </semantics></math> rounds by accepting <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mi>o</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mi>d</mi>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 2","pages":"293-312"},"PeriodicalIF":0.9,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142868961","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Kevin Hendrey, Sergey Norin, Raphael Steiner, Jérémie Turcotte
<p>Motivated by Hadwiger's conjecture, we study the problem of finding the densest possible <span></span><math> <mrow> <mi>t</mi> </mrow></math>-vertex minor in graphs of average degree at least <span></span><math> <mrow> <mi>t</mi> <mo>−</mo> <mn>1</mn> </mrow></math>. We show that if <span></span><math> <mrow> <mi>G</mi> </mrow></math> has average degree at least <span></span><math> <mrow> <mi>t</mi> <mo>−</mo> <mn>1</mn> </mrow></math>, it contains a minor on <span></span><math> <mrow> <mi>t</mi> </mrow></math> vertices with at least <span></span><math> <mrow> <mrow> <mo>(</mo> <mrow> <msqrt> <mn>2</mn> </msqrt> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>o</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mfenced> <mfrac> <mi>t</mi> <mn>2</mn> </mfrac> </mfenced> </mrow></math> edges. We show that this cannot be improved beyond <span></span><math> <mrow> <mfenced> <mrow> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> <mo>+</mo> <mi>o</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mfenced> <mfenced> <mfrac> <mi>t</mi> <mn>2</mn> </mfrac> </mfenced> </mrow></math>. Finally, for <span></span><math> <mrow> <mi>t</mi>
受 Hadwiger 猜想的启发，我们研究了在平均度至少为 t - 1 的图中寻找最密集的 t 个顶点次要顶点的问题。我们证明，如果 G 的平均度至少为 t - 1，那么它包含了 t 个顶点上的 minor，其中至少有 ( 2 - 1 - o ( 1 ) t 2 条边。我们证明这一点不能超过 3 4 + o ( 1 ) t 2 。最后，对于 t ≤ 6，我们精确地确定了在平均阶数至少为 t - 1 的图中，我们保证能在最密集的 t 个顶点次要图中找到的边的数量。
{"title":"Finding dense minors using average degree","authors":"Kevin Hendrey, Sergey Norin, Raphael Steiner, Jérémie Turcotte","doi":"10.1002/jgt.23169","DOIUrl":"https://doi.org/10.1002/jgt.23169","url":null,"abstract":"<p>Motivated by Hadwiger's conjecture, we study the problem of finding the densest possible <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow></math>-vertex minor in graphs of average degree at least <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>t</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow></math>. We show that if <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> has average degree at least <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>t</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow></math>, it contains a minor on <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow></math> vertices with at least <span></span><math>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <msqrt>\u0000 <mn>2</mn>\u0000 </msqrt>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mi>o</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mfenced>\u0000 <mfrac>\u0000 <mi>t</mi>\u0000 \u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </mfenced>\u0000 </mrow></math> edges. We show that this cannot be improved beyond <span></span><math>\u0000 \u0000 <mrow>\u0000 <mfenced>\u0000 <mrow>\u0000 <mfrac>\u0000 <mn>3</mn>\u0000 \u0000 <mn>4</mn>\u0000 </mfrac>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mi>o</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mfenced>\u0000 \u0000 <mfenced>\u0000 <mfrac>\u0000 <mi>t</mi>\u0000 \u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </mfenced>\u0000 </mrow></math>. Finally, for <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>t</mi>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 1","pages":"205-223"},"PeriodicalIF":0.9,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142707851","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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