Partial cubes are graphs that can be isometrically embedded into hypercubes. Convex cycles play an important role in the study of partial cubes. In this paper, we prove that a regular partial cube is a hypercube (resp., a Doubled Odd graph, an even cycle of length where ) if and only if all its convex cycles are 4-cycles (resp., 6-cycles, -cycles). In particular, the partial cubes whose all convex cycles are 4-cycles are equivalent to almost-median graphs. Therefore, we conclude that regular almost-median graphs are exactly hypercubes, which generalizes the result by Mulder—regular median graphs are hypercubes.
{"title":"A characterization of regular partial cubes whose all convex cycles have the same lengths","authors":"Yan-Ting Xie, Yong-De Feng, Shou-Jun Xu","doi":"10.1002/jgt.23126","DOIUrl":"10.1002/jgt.23126","url":null,"abstract":"<p>Partial cubes are graphs that can be isometrically embedded into hypercubes. Convex cycles play an important role in the study of partial cubes. In this paper, we prove that a regular partial cube is a hypercube (resp., a Doubled Odd graph, an even cycle of length <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $2n$</annotation>\u0000 </semantics></math> where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>⩾</mo>\u0000 \u0000 <mn>4</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $ngeqslant 4$</annotation>\u0000 </semantics></math>) if and only if all its convex cycles are 4-cycles (resp., 6-cycles, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $2n$</annotation>\u0000 </semantics></math>-cycles). In particular, the partial cubes whose all convex cycles are 4-cycles are equivalent to almost-median graphs. Therefore, we conclude that regular almost-median graphs are exactly hypercubes, which generalizes the result by Mulder—regular median graphs are hypercubes.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"550-558"},"PeriodicalIF":0.9,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504322","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>We derive attainable upper bounds on the algebraic connectivity (spectral gap) of a regular graph in terms of its diameter and girth. This bound agrees with the well-known Alon–Boppana–Friedman bound for graphs of even diameter, but is an improvement for graphs of odd diameter. For the girth bound, we show that only Moore graphs can attain it, and these only exist for well-known special cases. For the diameter bound, we use a combination of stochastic algorithms and exhaustive search to find graphs which attain it. For 3-regular graphs, we find attainable graphs for all diameters <span></span><math> <semantics> <mrow> <mrow> <mi>D</mi> </mrow> </mrow> <annotation> $D$</annotation> </semantics></math> up to and including <span></span><math> <semantics> <mrow> <mrow> <mi>D</mi> <mo>=</mo> <mn>9</mn> </mrow> </mrow> <annotation> $D=9$</annotation> </semantics></math> (the case of <span></span><math> <semantics> <mrow> <mrow> <mi>D</mi> <mo>=</mo> <mn>10</mn> </mrow> </mrow> <annotation> $D=10$</annotation> </semantics></math> is open). These graphs are extremely rare and also have high girth; for example, we found exactly 45 distinct cubic graphs on 44 vertices attaining the upper bound when <span></span><math> <semantics> <mrow> <mrow> <mi>D</mi> <mo>=</mo> <mn>7</mn> </mrow> </mrow> <annotation> $D=7$</annotation> </semantics></math>; all have girth 8. We also exhibit several infinite families attaining the upper bound with respect to diameter or girth. In particular, when <span></span><math> <semantics> <mrow> <mrow> <mi>d</mi> </mrow> </mrow> <annotation> $d$</annotation> </semantics></math> is a power of prime, we construct a <span></span><math> <semantics> <mrow> <mrow> <mi>d</mi> </mrow>
{"title":"Attainable bounds for algebraic connectivity and maximally connected regular graphs","authors":"Geoffrey Exoo, Theodore Kolokolnikov, Jeanette Janssen, Timothy Salamon","doi":"10.1002/jgt.23146","DOIUrl":"10.1002/jgt.23146","url":null,"abstract":"<p>We derive attainable upper bounds on the algebraic connectivity (spectral gap) of a regular graph in terms of its diameter and girth. This bound agrees with the well-known Alon–Boppana–Friedman bound for graphs of even diameter, but is an improvement for graphs of odd diameter. For the girth bound, we show that only Moore graphs can attain it, and these only exist for well-known special cases. For the diameter bound, we use a combination of stochastic algorithms and exhaustive search to find graphs which attain it. For 3-regular graphs, we find attainable graphs for all diameters <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $D$</annotation>\u0000 </semantics></math> up to and including <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>9</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $D=9$</annotation>\u0000 </semantics></math> (the case of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>10</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $D=10$</annotation>\u0000 </semantics></math> is open). These graphs are extremely rare and also have high girth; for example, we found exactly 45 distinct cubic graphs on 44 vertices attaining the upper bound when <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>7</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $D=7$</annotation>\u0000 </semantics></math>; all have girth 8. We also exhibit several infinite families attaining the upper bound with respect to diameter or girth. In particular, when <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $d$</annotation>\u0000 </semantics></math> is a power of prime, we construct a <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"522-549"},"PeriodicalIF":0.9,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23146","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504369","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>A finite group <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math> admits an <i>oriented regular representation</i> if there exists a Cayley digraph of <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math> such that it has no digons and its automorphism group is isomorphic to <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math>. Let <span></span><math> <semantics> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <annotation> $m$</annotation> </semantics></math> be a positive integer. In this paper, we extend the notion of oriented regular representations to oriented <span></span><math> <semantics> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <annotation> $m$</annotation> </semantics></math>-semiregular representations using <span></span><math> <semantics> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <annotation> $m$</annotation> </semantics></math>-Cayley digraphs. Given a finite group <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math>, an <span></span><math> <semantics> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <annotation> $m$</annotation> </semantics></math>-<i>Cayley digraph</i> of <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math> is a digraph that has a group of automorphisms isomorphic to <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi
{"title":"On oriented \u0000 \u0000 \u0000 \u0000 m\u0000 \u0000 \u0000 $m$\u0000 -semiregular representations of finite groups","authors":"Jia-Li Du, Yan-Quan Feng, Sejeong Bang","doi":"10.1002/jgt.23145","DOIUrl":"10.1002/jgt.23145","url":null,"abstract":"<p>A finite group <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> admits an <i>oriented regular representation</i> if there exists a Cayley digraph of <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> such that it has no digons and its automorphism group is isomorphic to <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>. Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math> be a positive integer. In this paper, we extend the notion of oriented regular representations to oriented <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math>-semiregular representations using <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math>-Cayley digraphs. Given a finite group <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>, an <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math>-<i>Cayley digraph</i> of <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> is a digraph that has a group of automorphisms isomorphic to <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"485-508"},"PeriodicalIF":0.9,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504370","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}
作为推论，我们证明对于任何阶数为 m $m$ 的树 T $T$ ，每一个 k $k$ -connected graph G $G$ δ ( G ) ≥ 3 k + 4 m - 6 $delta (G)ge 3k+4m-6$ 都包含一个子树 T ′ ≅ T ${T}^{^{prime} }cong T$，使得 G - V ( T ′ ) $G-V({T}^{^{prime} })$ 仍然是 k $k$ -connected 的，从而将马德的条件改进为线性约束。
<p>In this paper we study the average order of dominating sets in a graph, <span></span><math> <semantics> <mrow> <mrow> <mstyle> <mspace></mspace> <mtext>avd</mtext> <mspace></mspace> </mstyle> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </mrow> <annotation> $,text{avd},(G)$</annotation> </semantics></math>. Like other average graph parameters, the extremal graphs are of interest. Beaton and Brown conjectured that for all graphs <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math> of order <span></span><math> <semantics> <mrow> <mrow> <mi>n</mi> </mrow> </mrow> <annotation> $n$</annotation> </semantics></math> without isolated vertices, <span></span><math> <semantics> <mrow> <mrow> <mspace></mspace> <mtext>avd</mtext> <mspace></mspace> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>≤</mo> <mn>2</mn> <mi>n</mi> <mo>/</mo> <mn>3</mn> </mrow> </mrow> <annotation> $,text{avd},(G)le 2n/3$</annotation> </semantics></math>. Recently, Erey proved the conjecture for forests without isolated vertices. In this paper we prove the conjecture and classify which graphs have <span></span><math> <semantics> <mrow> <mrow> <mspace></mspace> <mtext>avd</mtext> <mspace></mspace>
本文研究图中支配集的平均阶数 avd ( G ) $,text{avd},(G)$ 。与其他平均图参数一样，极值图也很值得关注。比顿和布朗猜想，对于所有阶数为 n $n$ 的无孤立顶点的图 G $G$ ，avd ( G ) ≤ 2 n / 3 $,text{avd},(G)le 2n/3$ 。最近，埃雷证明了无孤立顶点森林的猜想。在本文中，我们证明了这个猜想，并分类了哪些图具有 avd ( G ) = 2 n / 3 $,text{avd},(G)=2n/3$ 。我们还利用我们的边界证明了平均版本的 Vizing 猜想。
{"title":"A tight upper bound on the average order of dominating sets of a graph","authors":"Iain Beaton, Ben Cameron","doi":"10.1002/jgt.23143","DOIUrl":"https://doi.org/10.1002/jgt.23143","url":null,"abstract":"<p>In this paper we study the average order of dominating sets in a graph, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mstyle>\u0000 <mspace></mspace>\u0000 \u0000 <mtext>avd</mtext>\u0000 <mspace></mspace>\u0000 </mstyle>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $,text{avd},(G)$</annotation>\u0000 </semantics></math>. Like other average graph parameters, the extremal graphs are of interest. Beaton and Brown conjectured that for all graphs <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> of order <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> without isolated vertices, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mspace></mspace>\u0000 \u0000 <mtext>avd</mtext>\u0000 <mspace></mspace>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mn>2</mn>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>/</mo>\u0000 \u0000 <mn>3</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $,text{avd},(G)le 2n/3$</annotation>\u0000 </semantics></math>. Recently, Erey proved the conjecture for forests without isolated vertices. In this paper we prove the conjecture and classify which graphs have <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mspace></mspace>\u0000 \u0000 <mtext>avd</mtext>\u0000 <mspace></mspace>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"463-477"},"PeriodicalIF":0.9,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23143","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142234963","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}
For a -uniform hypergraph we consider the parameter , the minimum size of a clique cover of the edge set of . We derive bounds on for belonging to various classes of hypergraphs.
对于一个 k $k$ 的均匀超图 F $F$ ，我们考虑参数 Θ ( F ) ${rm{Theta }}(F)$ ，即 F $F$ 边集的簇覆盖的最小大小。我们推导出属于各种超图类别的 F $F$ 的 Θ ( F ) ${rm{Theta }}(F)$ 的边界。
{"title":"Some results and problems on clique coverings of hypergraphs","authors":"Vojtech Rödl, Marcelo Sales","doi":"10.1002/jgt.23111","DOIUrl":"https://doi.org/10.1002/jgt.23111","url":null,"abstract":"<p>For a <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-uniform hypergraph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 <annotation> $F$</annotation>\u0000 </semantics></math> we consider the parameter <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Θ</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>F</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{Theta }}(F)$</annotation>\u0000 </semantics></math>, the minimum size of a clique cover of the edge set of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 <annotation> $F$</annotation>\u0000 </semantics></math>. We derive bounds on <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Θ</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>F</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{Theta }}(F)$</annotation>\u0000 </semantics></math> for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 <annotation> $F$</annotation>\u0000 </semantics></math> belonging to various classes of hypergraphs.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 2","pages":"442-457"},"PeriodicalIF":0.9,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23111","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141967877","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>Gyárfás and Sumner independently conjectured that for every tree <span></span><math> <semantics> <mrow> <mi>T</mi> </mrow> <annotation> $T$</annotation> </semantics></math>, there exists a function <span></span><math> <semantics> <mrow> <msub> <mi>f</mi> <mi>T</mi> </msub> <mo>:</mo> <mi>N</mi> <mo>→</mo> <mi>N</mi> </mrow> <annotation> ${f}_{T}:{mathbb{N}}to {mathbb{N}}$</annotation> </semantics></math> such that every <span></span><math> <semantics> <mrow> <mi>T</mi> </mrow> <annotation> $T$</annotation> </semantics></math>-free graph <span></span><math> <semantics> <mrow> <mi>G</mi> </mrow> <annotation> $G$</annotation> </semantics></math> satisfies <span></span><math> <semantics> <mrow> <mi>χ</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>≤</mo> <msub> <mi>f</mi> <mi>T</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>ω</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> $chi (G)le {f}_{T}(omega (G))$</annotation> </semantics></math>, where <span></span><math> <semantics> <mrow> <mi>χ</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> <annotation> $chi (G)$</annotation> </semantics></math> and <span></span><math> <semantics> <mrow> <mi>ω</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo>
Gyárfás 和 Sumner 独立猜想，对于每一棵树 T $T$ ，存在一个函数 f T : N → N ${f}_{T}:{mathbb{N}}to {mathbb{N}}$ ，使得每一个 T $T$ -free graph G $G$ 满足 χ ( G ) ≤ f T ( ω ( G ) ) $chi (G)le {f}_{T}(omega (G))$ ，其中 χ ( G ) $chi (G)$ 和 ω ( G ) $omega (G)$ 分别是 G $G$ 的色度数和小群数。这个猜想给出了关于色度数的拉姆齐式问题的解。对于图 G $G$，G $G$的诱导 SP-cover 数 inspc ( G ) $text{inspc}(G)$ （或者诱导 SP-partition 数 inspp ( G ) $text{inspp}(G)$ ）是 G $G$ 的诱导子图的族 P ${mathscr{P}}$ 的最小卡片度，使得 P ${mathscr{P}}$ 的每个元素都是星或路径，并且⋃ P ∈ P V ( P ) = V ( G ) ${bigcup }_{Pin {mathscr{P}}}V(P)=V(G)$ （或者诱导 SP-partition 数 inspp ( G ) $text{inspp}(G)$ ）是 G $G$ 的诱导子图的最小卡片度。
{"title":"Ramsey-type problems on induced covers and induced partitions toward the Gyárfás–Sumner conjecture","authors":"Shuya Chiba, Michitaka Furuya","doi":"10.1002/jgt.23124","DOIUrl":"https://doi.org/10.1002/jgt.23124","url":null,"abstract":"<p>Gyárfás and Sumner independently conjectured that for every tree <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math>, there exists a function <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>f</mi>\u0000 \u0000 <mi>T</mi>\u0000 </msub>\u0000 \u0000 <mo>:</mo>\u0000 \u0000 <mi>N</mi>\u0000 \u0000 <mo>→</mo>\u0000 \u0000 <mi>N</mi>\u0000 </mrow>\u0000 <annotation> ${f}_{T}:{mathbb{N}}to {mathbb{N}}$</annotation>\u0000 </semantics></math> such that every <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math>-free graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> satisfies <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>χ</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <msub>\u0000 <mi>f</mi>\u0000 \u0000 <mi>T</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>ω</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $chi (G)le {f}_{T}(omega (G))$</annotation>\u0000 </semantics></math>, where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>χ</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $chi (G)$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ω</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 2","pages":"419-441"},"PeriodicalIF":0.9,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141967063","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 <i>Kempe chain</i> on colors <span></span><math> <semantics> <mrow> <mi>a</mi> </mrow> <annotation> $a$</annotation> </semantics></math> and <span></span><math> <semantics> <mrow> <mi>b</mi> </mrow> <annotation> $b$</annotation> </semantics></math> is a component of the subgraph induced by colors <span></span><math> <semantics> <mrow> <mi>a</mi> </mrow> <annotation> $a$</annotation> </semantics></math> and <span></span><math> <semantics> <mrow> <mi>b</mi> </mrow> <annotation> $b$</annotation> </semantics></math>. A <i>Kempe change</i> is the operation of interchanging the colors of some Kempe chains. For a list-assignment <span></span><math> <semantics> <mrow> <mi>L</mi> </mrow> <annotation> $L$</annotation> </semantics></math> and an <span></span><math> <semantics> <mrow> <mi>L</mi> </mrow> <annotation> $L$</annotation> </semantics></math>-coloring <span></span><math> <semantics> <mrow> <mi>φ</mi> </mrow> <annotation> $varphi $</annotation> </semantics></math>, a Kempe change is <span></span><math> <semantics> <mrow> <mi>L</mi> </mrow> <annotation> $L$</annotation> </semantics></math>-<i>valid</i> for <span></span><math> <semantics> <mrow> <mi>φ</mi> </mrow> <annotation> $varphi $</annotation> </semantics></math> if performing the Kempe change yields another <span></span><math> <semantics> <mrow> <mi>L</mi> </mrow> <annotation> $L$</annotation> </semantics></math>-coloring. Two <span></span><math> <semantics> <mrow> <mi>L</mi> </mrow> <annotation> $L$</annotation> </semantics></math>-colorings are <span></span><math> <semantics> <mrow> <mi>L</mi> </mrow> <annotation> $L$</annotation> </semantics></math>-<i>equivalent</i> if we can form one from the other by a sequence of <span></span><math>
颜色 a $a$ 和 b $b$ 上的 Kempe 链是颜色 a $a$ 和 b $b$ 诱导的子图的一个组成部分。Kempe 变化是交换某些 Kempe 链颜色的操作。对于一个列表分配 L $L$ 和一个 L $L$ 颜色 φ $varphi $，如果进行 Kempe 更改能得到另一个 L $L$ 颜色，则 Kempe 更改对 φ $varphi $ 是 L $L$ 有效的。如果我们可以通过一连串 L $L$ 有效的 Kempe 变换从另一个 L $L$ 着色中得到一个 L $L$ 着色，那么这两个 L $L$ 着色就是 L $L$ 等价的。度赋值是一个列表赋值 L $L$，对于每个 v∈ V ( G ) $vin V(G)$ 来说，L ( v ) ≥ d ( v ) $L(v)ge d(v)$ 。克兰斯顿和马哈茂德问对于哪些图 G $G$ 和 G $G$ 的度数赋值 L $L$ 来说，G $G$ 的所有 L $L$ -着色都是 L $L$ -等价的？我们证明，对于每一个不完整的四连图 G $G$ 和 G $G$ 的每一个度数分配 L $L$, G $G$ 的所有 L $L$ -着色都是 L $L$ -等价的。
{"title":"Kempe equivalent list colorings revisited","authors":"Dibyayan Chakraborty, Carl Feghali, Reem Mahmoud","doi":"10.1002/jgt.23142","DOIUrl":"https://doi.org/10.1002/jgt.23142","url":null,"abstract":"<p>A <i>Kempe chain</i> on colors <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> is a component of the subgraph induced by colors <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>. A <i>Kempe change</i> is the operation of interchanging the colors of some Kempe chains. For a list-assignment <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math> and an <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math>-coloring <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>φ</mi>\u0000 </mrow>\u0000 <annotation> $varphi $</annotation>\u0000 </semantics></math>, a Kempe change is <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math>-<i>valid</i> for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>φ</mi>\u0000 </mrow>\u0000 <annotation> $varphi $</annotation>\u0000 </semantics></math> if performing the Kempe change yields another <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math>-coloring. Two <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math>-colorings are <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math>-<i>equivalent</i> if we can form one from the other by a sequence of <span></span><math>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 2","pages":"410-418"},"PeriodicalIF":0.9,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23142","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141966576","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>For a <span></span><math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math>-tree <span></span><math> <semantics> <mrow> <mi>T</mi> </mrow> <annotation> $T$</annotation> </semantics></math>, a generalization of a tree, the local mean order of sub-<span></span><math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math>-trees of <span></span><math> <semantics> <mrow> <mi>T</mi> </mrow> <annotation> $T$</annotation> </semantics></math> is the average order of sub-<span></span><math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math>-trees of <span></span><math> <semantics> <mrow> <mi>T</mi> </mrow> <annotation> $T$</annotation> </semantics></math> containing a given <span></span><math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math>-clique. The problem whether the maximum local mean order of a tree (i.e., a 1-tree) at a vertex is always taken on at a leaf was asked by Jamison in 1984 and was answered by Wagner and Wang in 2016. Actually, they proved that the maximum local mean order of a tree at a vertex occurs either at a leaf or at a vertex of degree 2. In 2018, Stephens and Oellermann asked a similar problem: for any <span></span><math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math>-tree <span></span><math> <semantics> <mrow> <mi>T</mi> </mrow> <annotation> $T$</annotation> </semantics></math>, does the maximum local mean order of sub-<span></span><math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math>-trees containing a given <span></span><math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math>-clique occur at a <span></span><math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math>-clique that is not a major <span></span
{"title":"On the maximum local mean order of sub-\u0000 \u0000 \u0000 k\u0000 \u0000 $k$\u0000 -trees of a \u0000 \u0000 \u0000 k\u0000 \u0000 $k$\u0000 -tree","authors":"Zhuo Li, Tianlong Ma, Fengming Dong, Xian'an Jin","doi":"10.1002/jgt.23128","DOIUrl":"10.1002/jgt.23128","url":null,"abstract":"<p>For a <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-tree <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math>, a generalization of a tree, the local mean order of sub-<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-trees of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math> is the average order of sub-<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-trees of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math> containing a given <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-clique. The problem whether the maximum local mean order of a tree (i.e., a 1-tree) at a vertex is always taken on at a leaf was asked by Jamison in 1984 and was answered by Wagner and Wang in 2016. Actually, they proved that the maximum local mean order of a tree at a vertex occurs either at a leaf or at a vertex of degree 2. In 2018, Stephens and Oellermann asked a similar problem: for any <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-tree <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math>, does the maximum local mean order of sub-<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-trees containing a given <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-clique occur at a <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-clique that is not a major <span></span","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 2","pages":"393-409"},"PeriodicalIF":0.9,"publicationDate":"2024-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141273372","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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