Suppose that the edges of a complete graph are assigned weights independently at random and we ask for the weight of the minimal-weight spanning tree, or perfect matching, or Hamiltonian cycle. For these and several other common optimisation problems, we establish asymptotically tight bounds when the weights are independent copies of a symmetric random variable (satisfying a mild condition on tail probabilities), in particular when the weights are Gaussian.
{"title":"Typical Values of Extremal-Weight Combinatorial Structures with Independent Symmetric Weights","authors":"Yun Cheng, Yixue Liu, T. Tkocz, Albert Xu","doi":"10.37236/10237","DOIUrl":"https://doi.org/10.37236/10237","url":null,"abstract":"Suppose that the edges of a complete graph are assigned weights independently at random and we ask for the weight of the minimal-weight spanning tree, or perfect matching, or Hamiltonian cycle. For these and several other common optimisation problems, we establish asymptotically tight bounds when the weights are independent copies of a symmetric random variable (satisfying a mild condition on tail probabilities), in particular when the weights are Gaussian.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"34 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76326537","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Pekárek, Jean-Sébastien Sereni, Zelealem B. Yilma
In a recent article, Bogdanowicz determines the minimum number of spanning trees a connected cubic multigraph on a fixed number of vertices can have and identifies the unique graph that attains this minimum value. He conjectures that a generalized form of this construction, which we here call a padded paddle graph, would be extremal for d-regular multigraphs where $dgeq 5$ is odd. We prove that, indeed, the padded paddle minimises the number of spanning trees, but this is true only when the number of vertices, $n$, is greater than $(9d+6)/8$. We show that a different graph, which we here call the padded cycle, is optimal for $n<(9d+6)/8$ . This fully determines the $d$-regular multi-graphs minimising the number of spanning trees for odd values of $d$. We employ the approach we develop to also consider and completely solve the even degree case. Here, the parity of $n$ plays a major role and we show that, apart from a handful of irregular cases when both $d$ and $n$ are small, the unique extremal graphs are padded cycles when $n$ is even and a different family, which we call fish graphs, when $n$ is odd.
{"title":"The Minimum Number of Spanning Trees in Regular Multigraphs","authors":"J. Pekárek, Jean-Sébastien Sereni, Zelealem B. Yilma","doi":"10.37236/10911","DOIUrl":"https://doi.org/10.37236/10911","url":null,"abstract":"\u0000 \u0000 \u0000In a recent article, Bogdanowicz determines the minimum number of spanning trees a connected cubic multigraph on a fixed number of vertices can have and identifies the unique graph that attains this minimum value. He conjectures that a generalized form of this construction, which we here call a padded paddle graph, would be extremal for d-regular multigraphs where $dgeq 5$ is odd. \u0000We prove that, indeed, the padded paddle minimises the number of spanning trees, but this is true only when the number of vertices, $n$, is greater than $(9d+6)/8$. We show that a different graph, which we here call the padded cycle, is optimal for $n<(9d+6)/8$ . This fully determines the $d$-regular multi-graphs minimising the number of spanning trees for odd values of $d$. \u0000We employ the approach we develop to also consider and completely solve the even degree case. Here, the parity of $n$ plays a major role and we show that, apart from a handful of irregular cases when both $d$ and $n$ are small, the unique extremal graphs are padded cycles when $n$ is even and a different family, which we call fish graphs, when $n$ is odd. \u0000 \u0000 \u0000","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"39 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75198044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a field $mathbb{F}$ and functions $f,g,h,jcolonmathbb{F}to mathbb{F}$, we define $Gamma_mathbb{F}(f(X)h(Y),g(X) j(Y))$ to be a bipartite graph where each partite set is a copy of $mathbb{F}^3$, and a vertex $(a,a_2,a_3)$ in the first partite set is adjacent to a vertex $[x,x_2,x_3]$ in the second partite set if and only if [a_2+x_2=f(a)h(x) quad text{and} quad a_3+x_3=g(a)j(x).] In this paper, we completely classify all such graphs by girth in the case $h=j$ (subject to some mild restrictions on $h$). We also present a partial classification when $hneq j$ and provide some applications.
{"title":"On the Girth of Three-Dimensional Algebraically Defined Graphs with Multiplicatively Separable Functions","authors":"Alex Kodess, Brian G. Kronenthal, Tony W. H. Wong","doi":"10.37236/9749","DOIUrl":"https://doi.org/10.37236/9749","url":null,"abstract":"For a field $mathbb{F}$ and functions $f,g,h,jcolonmathbb{F}to mathbb{F}$, we define $Gamma_mathbb{F}(f(X)h(Y),g(X) j(Y))$ to be a bipartite graph where each partite set is a copy of $mathbb{F}^3$, and a vertex $(a,a_2,a_3)$ in the first partite set is adjacent to a vertex $[x,x_2,x_3]$ in the second partite set if and only if [a_2+x_2=f(a)h(x) quad text{and} quad a_3+x_3=g(a)j(x).] In this paper, we completely classify all such graphs by girth in the case $h=j$ (subject to some mild restrictions on $h$). We also present a partial classification when $hneq j$ and provide some applications.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"3 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83741758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Irfan Durmi'c, A. Han, P. Harris, Rodrigo Ribeiro, Mei Yin
We consider the notion of classical parking functions by introducing randomness and a new parking protocol, as inspired by the work presented in the paper ``Parking Functions: Choose your own adventure,'' (arXiv:2001.04817) by Carlson, Christensen, Harris, Jones, and Rodríguez. Among our results, we prove that the probability of obtaining a parking function, from a length $n$ preference vector, is independent of the probabilistic parameter $p$. We also explore the properties of a preference vector given that it is a parking function and discuss the effect of the probabilistic parameter $p$. Of special interest is when $p=1/2$, where we demonstrate a sharp transition in some parking statistics. We also present several interesting combinatorial consequences of the parking protocol. In particular, we provide a combinatorial interpretation for the array described in OEIS A220884 as the expected number of preference sequences with a particular property related to occupied parking spots. Lastly, we connect our results to other weighted phenomena in combinatorics and provide further directions for research.
{"title":"Probabilistic Parking Functions","authors":"Irfan Durmi'c, A. Han, P. Harris, Rodrigo Ribeiro, Mei Yin","doi":"10.37236/11649","DOIUrl":"https://doi.org/10.37236/11649","url":null,"abstract":"We consider the notion of classical parking functions by introducing randomness and a new parking protocol, as inspired by the work presented in the paper ``Parking Functions: Choose your own adventure,'' (arXiv:2001.04817) by Carlson, Christensen, Harris, Jones, and Rodríguez. Among our results, we prove that the probability of obtaining a parking function, from a length $n$ preference vector, is independent of the probabilistic parameter $p$. We also explore the properties of a preference vector given that it is a parking function and discuss the effect of the probabilistic parameter $p$. Of special interest is when $p=1/2$, where we demonstrate a sharp transition in some parking statistics. We also present several interesting combinatorial consequences of the parking protocol. In particular, we provide a combinatorial interpretation for the array described in OEIS A220884 as the expected number of preference sequences with a particular property related to occupied parking spots. Lastly, we connect our results to other weighted phenomena in combinatorics and provide further directions for research.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"27 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74058618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A weakly distance-regular digraph is $P$-polynomial if its attached scheme is $P$-polynomial. In this paper, we characterize all $P$-polynomial weakly distance-regular digraphs.
{"title":"$P$-Polynomial Weakly Distance-Regular Digraphs","authors":"Zeng Qing, Yuefeng Yang, Kaishun Wang","doi":"10.37236/11798","DOIUrl":"https://doi.org/10.37236/11798","url":null,"abstract":"A weakly distance-regular digraph is $P$-polynomial if its attached scheme is $P$-polynomial. In this paper, we characterize all $P$-polynomial weakly distance-regular digraphs.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"50 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89165285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A new descent set statistic on involutions, defined geometrically via their interpretation as matchings, is introduced in this paper, and shown to be equidistributed with the standard one. This concept is then applied to construct explicit cyclic descent extensions on involutions, standard Young tableaux and Motzkin paths. Schur-positivity of the associated quasisymmetric functions follows.
{"title":"Cyclic Descents, Matchings and Schur-Positivity","authors":"R. Adin, Yuval Roichman","doi":"10.37236/11761","DOIUrl":"https://doi.org/10.37236/11761","url":null,"abstract":"A new descent set statistic on involutions, defined geometrically via their interpretation as matchings, is introduced in this paper, and shown to be equidistributed with the standard one. This concept is then applied to construct explicit cyclic descent extensions on involutions, standard Young tableaux and Motzkin paths. Schur-positivity of the associated quasisymmetric functions follows.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"109 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86785989","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For graphs $F$ and $H$, the Ramsey number $R(F, H)$ is the smallest positive integer $N$ such that any red/blue edge coloring of $K_N$ contains either a red $F$ or a blue $H$. Let $C_n$ be a cycle of length $n$ and $F_n$ be a fan consisting of $n$ triangles all sharing a common vertex.In this paper, we prove that for all sufficiently large $n$,[R(C_{2lfloor anrfloor}, F_n)= left{ begin{array}{ll}(2+2a+o(1))n & textrm{if $1/2leq a< 1$,}(4a+o(1))n & textrm{if $ ageq 1$.}end{array} right.]
{"title":"Ramsey Numbers of Large Even Cycles and Fans","authors":"Chunlin You, Qizhong Lin","doi":"10.37236/11068","DOIUrl":"https://doi.org/10.37236/11068","url":null,"abstract":"For graphs $F$ and $H$, the Ramsey number $R(F, H)$ is the smallest positive integer $N$ such that any red/blue edge coloring of $K_N$ contains either a red $F$ or a blue $H$. Let $C_n$ be a cycle of length $n$ and $F_n$ be a fan consisting of $n$ triangles all sharing a common vertex.In this paper, we prove that for all sufficiently large $n$,[R(C_{2lfloor anrfloor}, F_n)= left{ begin{array}{ll}(2+2a+o(1))n & textrm{if $1/2leq a< 1$,}(4a+o(1))n & textrm{if $ ageq 1$.}end{array} right.]","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"36 5","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72487224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Rutger Campbell, Marc Distel, J. P. Gollin, Daniel J. Harvey, Kevin Hendrey, Robert Hickingbotham, B. Mohar, D. Wood
A graph class $mathcal{G}$ has linear growth if, for each graph $G in mathcal{G}$ and every positive integer $r$, every subgraph of $G$ with radius at most $r$ contains $O(r)$ vertices. In this paper, we show that every graph class with linear growth has bounded treewidth.
一个图类$mathcal{G}$具有线性增长，如果对于每一个图$G in $ mathcal{G}$和每一个正整数$r$， $G$的每一个半径不超过$r$的子图包含$O(r)$顶点。本文证明了每一类具有线性增长的图都有有界树宽。
{"title":"Graphs of Linear Growth have Bounded Treewidth","authors":"Rutger Campbell, Marc Distel, J. P. Gollin, Daniel J. Harvey, Kevin Hendrey, Robert Hickingbotham, B. Mohar, D. Wood","doi":"10.37236/11657","DOIUrl":"https://doi.org/10.37236/11657","url":null,"abstract":"A graph class $mathcal{G}$ has linear growth if, for each graph $G in mathcal{G}$ and every positive integer $r$, every subgraph of $G$ with radius at most $r$ contains $O(r)$ vertices. In this paper, we show that every graph class with linear growth has bounded treewidth.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"18 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80516477","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The prism over a graph $G$ is the Cartesian product of $G$ with the complete graph $K_2$. The graph $G$ is prism-hamiltonian if the prism over $G$ has a Hamilton cycle. A good even cactus is a connected graph in which every block is either an edge or an even cycle and every vertex is contained in at most two blocks. It is known that good even cacti are prism-hamiltonian. Indeed, showing the existence of a spanning good even cactus has become the most common technique in proving prism-hamiltonicity. Špacapan [S. Špacapan. A counterexample to prism-hamiltonicity of 3-connected planar graphs. J. Combin. Theory Ser. B, 146:364--371, 2021] asked whether having a spanning good even cactus is equivalent to having a hamiltonian prism for 3-connected planar graphs. In this article we answer his question in the negative, by showing that there are infinitely many 3-connected planar prism-hamiltonian graphs that have no spanning good even cactus. In addition, we prove the existence of an infinite class of 3-connected planar graphs that have a spanning good even cactus but no spanning good even cactus with maximum degree three.
{"title":"On the Hamiltonian Property Hierarchy of 3-Connected Planar Graphs","authors":"O. S. Lo","doi":"10.37236/11070","DOIUrl":"https://doi.org/10.37236/11070","url":null,"abstract":"The prism over a graph $G$ is the Cartesian product of $G$ with the complete graph $K_2$. The graph $G$ is prism-hamiltonian if the prism over $G$ has a Hamilton cycle. A good even cactus is a connected graph in which every block is either an edge or an even cycle and every vertex is contained in at most two blocks. It is known that good even cacti are prism-hamiltonian. Indeed, showing the existence of a spanning good even cactus has become the most common technique in proving prism-hamiltonicity. Špacapan [S. Špacapan. A counterexample to prism-hamiltonicity of 3-connected planar graphs. J. Combin. Theory Ser. B, 146:364--371, 2021] asked whether having a spanning good even cactus is equivalent to having a hamiltonian prism for 3-connected planar graphs. In this article we answer his question in the negative, by showing that there are infinitely many 3-connected planar prism-hamiltonian graphs that have no spanning good even cactus. In addition, we prove the existence of an infinite class of 3-connected planar graphs that have a spanning good even cactus but no spanning good even cactus with maximum degree three.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"64 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80142960","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce the cyclic major index of a cyclic permutation and give a bivariate analogue of the enumerative formula for the cyclic shuffles with a given cyclic descent number due to Adin, Gessel, Reiner and Roichman, which can be viewed as a cyclic analogue of Stanley's shuffling theorem. This gives an answer to a question of Adin, Gessel, Reiner and Roichman, which has been posed by Domagalski, Liang, Minnich, Sagan, Schmidt and Sietsema again.
{"title":"A Cyclic Analogue of Stanley's Shuffling Theorem","authors":"K. Ji, Daxun Zhang","doi":"10.37236/11238","DOIUrl":"https://doi.org/10.37236/11238","url":null,"abstract":"We introduce the cyclic major index of a cyclic permutation and give a bivariate analogue of the enumerative formula for the cyclic shuffles with a given cyclic descent number due to Adin, Gessel, Reiner and Roichman, which can be viewed as a cyclic analogue of Stanley's shuffling theorem. This gives an answer to a question of Adin, Gessel, Reiner and Roichman, which has been posed by Domagalski, Liang, Minnich, Sagan, Schmidt and Sietsema again.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"17 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84420852","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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