Pub Date : 2025-10-30DOI: 10.1016/j.ejc.2025.104259
Rebecca Coulson
Metrically homogeneous graphs are connected graphs which, when endowed with the path metric, are homogeneous as metric spaces. In this paper we introduce the concept of twisted automorphisms, a notion of isomorphism up to a permutation of the language. We find all permutations of the language which are associated with twisted automorphisms of metrically homogeneous graphs. For each non-trivial permutation of this type we also characterize the class of metrically homogeneous graphs which allow a twisted isomorphism associated with that permutation. The permutations we find are, remarkably, precisely those found by Bannai and Bannai in an analogous result in the context of finite association schemes (Bannai and Bannai, 1980), though why this might be is still an open question.
{"title":"Twists and twistability","authors":"Rebecca Coulson","doi":"10.1016/j.ejc.2025.104259","DOIUrl":"10.1016/j.ejc.2025.104259","url":null,"abstract":"<div><div>Metrically homogeneous graphs are connected graphs which, when endowed with the path metric, are homogeneous as metric spaces. In this paper we introduce the concept of <em>twisted automorphisms</em>, a notion of isomorphism up to a permutation of the language. We find all permutations of the language which are associated with twisted automorphisms of metrically homogeneous graphs. For each non-trivial permutation of this type we also characterize the class of metrically homogeneous graphs which allow a twisted isomorphism associated with that permutation. The permutations we find are, remarkably, precisely those found by Bannai and Bannai in an analogous result in the context of finite association schemes (Bannai and Bannai, 1980), though why this might be is still an open question.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104259"},"PeriodicalIF":0.9,"publicationDate":"2025-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145623512","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}
Pub Date : 2025-10-27DOI: 10.1016/j.ejc.2025.104269
Yuhang Bai , Kristóf Bérczi , Gergely Csáji , Tamás Schwarcz
In the Properly Colored Spanning Tree problem, we are given an edge-colored undirected graph and the goal is to find a properly colored spanning tree, i.e., a spanning tree in which any two adjacent edges have distinct colors. The problem is interesting not only from a graph coloring point of view, but is also closely related to the Degree Bounded Spanning Tree and -Traveling Salesman problems, two classical questions that have attracted considerable interest in combinatorial optimization and approximation theory. Previous work on properly colored spanning trees has mainly focused on determining the existence of such a tree and hence has not considered the question from an algorithmic perspective. We propose an optimization version called Maximum-size Properly Colored Forest problem, which aims to find a properly colored forest with as many edges as possible. We consider the problem in different graph classes and for different numbers of colors, and present polynomial-time approximation algorithms as well as inapproximability results for these settings. Our proof technique relies on the sum of matching matroids defined by the color classes, a connection that might be of independent combinatorial interest. We also consider the Maximum-size Properly Colored Tree problem asking for the maximum size of a properly colored tree not necessarily spanning all the vertices. We show that the optimum is significantly more difficult to approximate than in the forest case, and provide an approximation algorithm for complete multigraphs.
{"title":"Approximating maximum-size properly colored forests","authors":"Yuhang Bai , Kristóf Bérczi , Gergely Csáji , Tamás Schwarcz","doi":"10.1016/j.ejc.2025.104269","DOIUrl":"10.1016/j.ejc.2025.104269","url":null,"abstract":"<div><div>In the <em>Properly Colored Spanning Tree</em> problem, we are given an edge-colored undirected graph and the goal is to find a properly colored spanning tree, i.e., a spanning tree in which any two adjacent edges have distinct colors. The problem is interesting not only from a graph coloring point of view, but is also closely related to the <em>Degree Bounded Spanning Tree</em> and <span><math><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></math></span>-<em>Traveling Salesman</em> problems, two classical questions that have attracted considerable interest in combinatorial optimization and approximation theory. Previous work on properly colored spanning trees has mainly focused on determining the existence of such a tree and hence has not considered the question from an algorithmic perspective. We propose an optimization version called <em>Maximum-size Properly Colored Forest</em> problem, which aims to find a properly colored forest with as many edges as possible. We consider the problem in different graph classes and for different numbers of colors, and present polynomial-time approximation algorithms as well as inapproximability results for these settings. Our proof technique relies on the sum of matching matroids defined by the color classes, a connection that might be of independent combinatorial interest. We also consider the <em>Maximum-size Properly Colored Tree</em> problem asking for the maximum size of a properly colored tree not necessarily spanning all the vertices. We show that the optimum is significantly more difficult to approximate than in the forest case, and provide an approximation algorithm for complete multigraphs.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104269"},"PeriodicalIF":0.9,"publicationDate":"2025-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418555","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}
Pub Date : 2025-10-27DOI: 10.1016/j.ejc.2025.104266
Pierre Simon
We establish new results on the possible growth rates for the sequence counting the number of orbits of a given oligomorphic group on unordered sets of size . Macpherson showed that for primitive actions, the growth is at least exponential (if the sequence is not constant equal to 1). The best lower bound previously known for the base of the exponential was obtained by Merola. We establishing the optimal value of 2 in the case where the structure is unstable. This allows us to improve on Merola’s bound and also obtain the optimal value for structures homogeneous in a finite relational language. Finally, we show that the study of sequences of sub-exponential growth reduces to the -stable case.
{"title":"On ω-categorical structures with few finite substructures","authors":"Pierre Simon","doi":"10.1016/j.ejc.2025.104266","DOIUrl":"10.1016/j.ejc.2025.104266","url":null,"abstract":"<div><div>We establish new results on the possible growth rates for the sequence <span><math><mrow><mo>(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></math></span> counting the number of orbits of a given oligomorphic group on unordered sets of size <span><math><mi>n</mi></math></span>. Macpherson showed that for primitive actions, the growth is at least exponential (if the sequence is not constant equal to 1). The best lower bound previously known for the base of the exponential was obtained by Merola. We establishing the optimal value of 2 in the case where the structure is unstable. This allows us to improve on Merola’s bound and also obtain the optimal value for structures homogeneous in a finite relational language. Finally, we show that the study of sequences <span><math><mrow><mo>(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></math></span> of sub-exponential growth reduces to the <span><math><mi>ω</mi></math></span>-stable case.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104266"},"PeriodicalIF":0.9,"publicationDate":"2025-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145624163","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}
Pub Date : 2025-10-21DOI: 10.1016/j.ejc.2025.104258
David Conlon , Nina Kamčev
The Hales–Jewett theorem states that for any and there exists an such that any -colouring of the elements of contains a monochromatic combinatorial line. We study the structure of the wildcard set which determines this monochromatic line, showing that when is odd there are -colourings of where the wildcard set of a monochromatic line cannot be the union of fewer than intervals. This is tight, as for sufficiently large there are always monochromatic lines whose wildcard set is the union of at most intervals.
{"title":"Intervals in the Hales–Jewett theorem","authors":"David Conlon , Nina Kamčev","doi":"10.1016/j.ejc.2025.104258","DOIUrl":"10.1016/j.ejc.2025.104258","url":null,"abstract":"<div><div>The Hales–Jewett theorem states that for any <span><math><mi>m</mi></math></span> and <span><math><mi>r</mi></math></span> there exists an <span><math><mi>n</mi></math></span> such that any <span><math><mi>r</mi></math></span>-colouring of the elements of <span><math><msup><mrow><mrow><mo>[</mo><mi>m</mi><mo>]</mo></mrow></mrow><mrow><mi>n</mi></mrow></msup></math></span> contains a monochromatic combinatorial line. We study the structure of the wildcard set <span><math><mrow><mi>S</mi><mo>⊆</mo><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow></math></span> which determines this monochromatic line, showing that when <span><math><mi>r</mi></math></span> is odd there are <span><math><mi>r</mi></math></span>-colourings of <span><math><msup><mrow><mrow><mo>[</mo><mn>3</mn><mo>]</mo></mrow></mrow><mrow><mi>n</mi></mrow></msup></math></span> where the wildcard set of a monochromatic line cannot be the union of fewer than <span><math><mi>r</mi></math></span> intervals. This is tight, as for <span><math><mi>n</mi></math></span> sufficiently large there are always monochromatic lines whose wildcard set is the union of at most <span><math><mi>r</mi></math></span> intervals.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104258"},"PeriodicalIF":0.9,"publicationDate":"2025-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145623511","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}
Pub Date : 2025-10-18DOI: 10.1016/j.ejc.2025.104256
Andrés Aranda , David Hartman
In Hartman et al. (2014), Hartman, Hubička and Mašulović studied the hierarchy of morphism extension classes for finite -colored graphs, that is, undirected graphs without loops where sets of colors selected from are assigned to vertices and edges. They proved that when is a linear order, the classes and coincide, and the same is true for vertex-uniform finite -colored graphs when is a diamond. In this paper, we explore the same question for countably infinite -colored graphs. We prove that if and only if is a linear order.
在Hartman et al.（2014）中，Hartman, hubi ka和Mašulović研究了有限L色图的态射扩展类的层次结构，即无环路的无向图，其中从L中选择的颜色集分配给顶点和边。他们证明了当L是线性阶时，类MHL和类HHL重合，当L是菱形时，同样适用于顶点一致的有限L色图。本文探讨了可数无限l色图的相同问题。证明MHL=HHL当且仅当L是线性阶。
{"title":"Morphism extension classes of countable L-colored graphs","authors":"Andrés Aranda , David Hartman","doi":"10.1016/j.ejc.2025.104256","DOIUrl":"10.1016/j.ejc.2025.104256","url":null,"abstract":"<div><div>In Hartman et al. (2014), Hartman, Hubička and Mašulović studied the hierarchy of morphism extension classes for finite <span><math><mi>L</mi></math></span>-colored graphs, that is, undirected graphs without loops where sets of colors selected from <span><math><mi>L</mi></math></span> are assigned to vertices and edges. They proved that when <span><math><mi>L</mi></math></span> is a linear order, the classes <span><math><msub><mrow><mi>MH</mi></mrow><mrow><mi>L</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>HH</mi></mrow><mrow><mi>L</mi></mrow></msub></math></span> coincide, and the same is true for vertex-uniform finite <span><math><mi>L</mi></math></span>-colored graphs when <span><math><mi>L</mi></math></span> is a diamond. In this paper, we explore the same question for countably infinite <span><math><mi>L</mi></math></span>-colored graphs. We prove that <span><math><mrow><msub><mrow><mi>MH</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>=</mo><msub><mrow><mi>HH</mi></mrow><mrow><mi>L</mi></mrow></msub></mrow></math></span> if and only if <span><math><mi>L</mi></math></span> is a linear order.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104256"},"PeriodicalIF":0.9,"publicationDate":"2025-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145623509","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}
Pub Date : 2025-10-18DOI: 10.1016/j.ejc.2025.104262
Jan Hubička , Michael Kompatscher , Matěj Konečný
In a recent paper by a superset of the authors it was proved that for every primitive 3-constrained space of finite diameter from Cherlin’s catalogue of metrically homogeneous graphs, there exists a finite family of -edge-labelled cycles such that a -edge-labelled graph is a subgraph of if and only if it contains no homomorphic images of cycles from . However, the cycles in the families were not described explicitly as it was not necessary for the analysis of Ramsey expansions and the extension property for partial automorphisms. This paper fills this gap by providing an explicit description of the cycles in the families , heavily using the previous result in the process. Additionally, we explore the potential applications of this result, such as interpreting the graphs as semigroup-valued metric spaces or homogenisations of -categorical -edge-labelled graphs.
{"title":"Forbidden cycles in metrically homogeneous graphs","authors":"Jan Hubička , Michael Kompatscher , Matěj Konečný","doi":"10.1016/j.ejc.2025.104262","DOIUrl":"10.1016/j.ejc.2025.104262","url":null,"abstract":"<div><div>In a recent paper by a superset of the authors it was proved that for every primitive 3-constrained space <span><math><mi>Γ</mi></math></span> of finite diameter <span><math><mi>δ</mi></math></span> from Cherlin’s catalogue of metrically homogeneous graphs, there exists a finite family <span><math><mi>F</mi></math></span> of <span><math><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>δ</mi><mo>}</mo></mrow></math></span>-edge-labelled cycles such that a <span><math><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>δ</mi><mo>}</mo></mrow></math></span>-edge-labelled graph is a subgraph of <span><math><mi>Γ</mi></math></span> if and only if it contains no homomorphic images of cycles from <span><math><mi>F</mi></math></span>. However, the cycles in the families <span><math><mi>F</mi></math></span> were not described explicitly as it was not necessary for the analysis of Ramsey expansions and the extension property for partial automorphisms. This paper fills this gap by providing an explicit description of the cycles in the families <span><math><mi>F</mi></math></span>, heavily using the previous result in the process. Additionally, we explore the potential applications of this result, such as interpreting the graphs as semigroup-valued metric spaces or homogenisations of <span><math><mi>ω</mi></math></span>-categorical <span><math><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mi>δ</mi><mo>}</mo></mrow></math></span>-edge-labelled graphs.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104262"},"PeriodicalIF":0.9,"publicationDate":"2025-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145623515","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}
Pub Date : 2025-10-18DOI: 10.1016/j.ejc.2025.104255
Andrés Aranda , David Bradley-Williams , Jan Hubička , Miltiadis Karamanlis , Michael Kompatscher , Matěj Konečný , Micheal Pawliuk
We investigate Ramsey expansions, the coherent extension property for partial isometries (EPPA), and the existence of a stationary independence relation for all classes of metrically homogeneous graphs from Cherlin’s catalogue. We show that, with the exception of tree-like graphs, all metric spaces in the catalogue have precompact Ramsey expansions (or lifts) with the expansion property. With two exceptions we can also characterise the existence of a stationary independence relation and coherent EPPA.
Our results are a contribution to Nešetřil’s classification programme of Ramsey classes and can be seen as empirical evidence of the recent convergence in techniques employed to establish the Ramsey property, the expansion property, EPPA and the existence of a stationary independence relation. At the heart of our proof is a canonical way of completing edge-labelled graphs to metric spaces in Cherlin’s classes. The existence of such a “completion algorithm” then allows us to apply several strong results in the areas that imply EPPA or the Ramsey property.
The main results have numerous consequences for the automorphism groups of the Fraïssé limits of the classes. As corollaries, we prove amenability, unique ergodicity, existence of universal minimal flows, ample generics, small index property, 21-Bergman property and Serre’s property (FA).
{"title":"Ramsey expansions of metrically homogeneous graphs","authors":"Andrés Aranda , David Bradley-Williams , Jan Hubička , Miltiadis Karamanlis , Michael Kompatscher , Matěj Konečný , Micheal Pawliuk","doi":"10.1016/j.ejc.2025.104255","DOIUrl":"10.1016/j.ejc.2025.104255","url":null,"abstract":"<div><div>We investigate Ramsey expansions, the coherent extension property for partial isometries (EPPA), and the existence of a stationary independence relation for all classes of metrically homogeneous graphs from Cherlin’s catalogue. We show that, with the exception of tree-like graphs, all metric spaces in the catalogue have precompact Ramsey expansions (or lifts) with the expansion property. With two exceptions we can also characterise the existence of a stationary independence relation and coherent EPPA.</div><div>Our results are a contribution to Nešetřil’s classification programme of Ramsey classes and can be seen as empirical evidence of the recent convergence in techniques employed to establish the Ramsey property, the expansion property, EPPA and the existence of a stationary independence relation. At the heart of our proof is a canonical way of completing edge-labelled graphs to metric spaces in Cherlin’s classes. The existence of such a “completion algorithm” then allows us to apply several strong results in the areas that imply EPPA or the Ramsey property.</div><div>The main results have numerous consequences for the automorphism groups of the Fraïssé limits of the classes. As corollaries, we prove amenability, unique ergodicity, existence of universal minimal flows, ample generics, small index property, 21-Bergman property and Serre’s property (FA).</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104255"},"PeriodicalIF":0.9,"publicationDate":"2025-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145623508","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}
Pub Date : 2025-10-16DOI: 10.1016/j.ejc.2025.104263
Colin Jahel , Andy Zucker
We consider a short exact sequence of Polish groups and consider what can be deduced about the dynamics of given information about the dynamics of and . We prove that if the respective universal minimal flows and are metrizable, then so is . Furthermore, we show that if and are metrizable and both and are uniquely ergodic, then so is . We then discuss several examples of these phenomena.
{"title":"Topological dynamics of Polish group extensions","authors":"Colin Jahel , Andy Zucker","doi":"10.1016/j.ejc.2025.104263","DOIUrl":"10.1016/j.ejc.2025.104263","url":null,"abstract":"<div><div>We consider a short exact sequence <span><math><mrow><mn>1</mn><mo>→</mo><mi>H</mi><mo>→</mo><mi>G</mi><mo>→</mo><mi>K</mi><mo>→</mo><mn>1</mn></mrow></math></span> of Polish groups and consider what can be deduced about the dynamics of <span><math><mi>G</mi></math></span> given information about the dynamics of <span><math><mi>H</mi></math></span> and <span><math><mi>K</mi></math></span>. We prove that if the respective universal minimal flows <span><math><mrow><mi>M</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>M</mi><mrow><mo>(</mo><mi>K</mi><mo>)</mo></mrow></mrow></math></span> are metrizable, then so is <span><math><mrow><mi>M</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Furthermore, we show that if <span><math><mrow><mi>M</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>M</mi><mrow><mo>(</mo><mi>K</mi><mo>)</mo></mrow></mrow></math></span> are metrizable and both <span><math><mi>H</mi></math></span> and <span><math><mi>K</mi></math></span> are uniquely ergodic, then so is <span><math><mi>G</mi></math></span>. We then discuss several examples of these phenomena.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104263"},"PeriodicalIF":0.9,"publicationDate":"2025-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145624160","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}
Pub Date : 2025-10-16DOI: 10.1016/j.ejc.2025.104267
Fredy Yip
Recently, Alon initiated the study of graph codes and their linear variants in analogy to the study of error correcting codes in theoretical computer science. Alon related the maximum density of a linear graph code which avoids images of a small graph to the following variant of the Erdős–Gyárfás problem on edge-colourings of . A copy of in an edge-colouring of is even-chromatic if each colour occupies an even number of edges in the copy. We seek an edge-colouring of using colours such that there are no even-chromatic copies of . Such an edge-colouring is conjectured to exist for all cliques with an even number of edges. To date, edge-colourings satisfying this property have been constructed for and .
We construct an edge-colouring using colours which avoids even-chromatic copies of . This was the smallest open case of the above conjecture, as each has an odd number of edges. We also study a stronger condition on edge-colourings, where for each copy of , there is a colour occupying exactly one edge in the copy. We conjecture that an edge-colouring using colours and satisfying this stronger requirement exists for all cliques regardless of the parity of the number of its edges. We construct edge-colourings satisfying this stronger property for and . These constructions also improve upon the number of colours needed for the original problem of avoiding even-chromatic copies of
{"title":"A variant of the Erdős–Gyárfás problem for K8","authors":"Fredy Yip","doi":"10.1016/j.ejc.2025.104267","DOIUrl":"10.1016/j.ejc.2025.104267","url":null,"abstract":"<div><div>Recently, Alon initiated the study of graph codes and their linear variants in analogy to the study of error correcting codes in theoretical computer science. Alon related the maximum density of a linear graph code which avoids images of a small graph <span><math><mi>H</mi></math></span> to the following variant of the Erdős–Gyárfás problem on edge-colourings of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. A copy of <span><math><mi>H</mi></math></span> in an edge-colouring of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is <em>even-chromatic</em> if each colour occupies an even number of edges in the copy. We seek an edge-colouring of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> using <span><math><msup><mrow><mi>n</mi></mrow><mrow><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup></math></span> colours such that there are no even-chromatic copies of <span><math><mi>H</mi></math></span>. Such an edge-colouring is conjectured to exist for all cliques <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> with an even number of edges. To date, edge-colourings satisfying this property have been constructed for <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>.</div><div>We construct an edge-colouring using <span><math><msup><mrow><mi>n</mi></mrow><mrow><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup></math></span> colours which avoids even-chromatic copies of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>8</mn></mrow></msub></math></span>. This was the smallest open case of the above conjecture, as <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>7</mn></mrow></msub></mrow></math></span> each has an odd number of edges. We also study a stronger condition on edge-colourings, where for each copy of <span><math><mi>H</mi></math></span>, there is a colour occupying exactly one edge in the copy. We conjecture that an edge-colouring using <span><math><msup><mrow><mi>n</mi></mrow><mrow><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup></math></span> colours and satisfying this stronger requirement exists for all cliques <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> regardless of the parity of the number of its edges. We construct edge-colourings satisfying this stronger property for <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>. These constructions also improve upon the number of colours needed for the original problem of avoiding even-chromatic copies of <span><math><msub","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104267"},"PeriodicalIF":0.9,"publicationDate":"2025-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145326854","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}
Pub Date : 2025-10-16DOI: 10.1016/j.ejc.2025.104249
Robert P. Laudone
We show that cyclic permutations avoiding 321 are precisely those permutations whose image under the fundamental bijection avoid a set of vincular patterns. We do this by using pattern functions and arrow patterns, in combination with the characterization of 321 avoidance in terms of equality of the upper bound of the Daiconis–Graham inequalities. We then explore some consequences of this result, including upper and lower bound results on the growth rate of 321 avoiding cycles.
{"title":"Characterizing avoidance in cycles via vincular patterns","authors":"Robert P. Laudone","doi":"10.1016/j.ejc.2025.104249","DOIUrl":"10.1016/j.ejc.2025.104249","url":null,"abstract":"<div><div>We show that cyclic permutations avoiding 321 are precisely those permutations whose image under the fundamental bijection avoid a set of vincular patterns. We do this by using pattern functions and arrow patterns, in combination with the characterization of 321 avoidance in terms of equality of the upper bound of the Daiconis–Graham inequalities. We then explore some consequences of this result, including upper and lower bound results on the growth rate of 321 avoiding cycles.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104249"},"PeriodicalIF":0.9,"publicationDate":"2025-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145326522","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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