Pub Date : 2021-08-06DOI: 10.26493/1855-3974.1955.1cd
César Hernández-Cruz, M. Petrusevski, R. Škrekovski
A weak-odd edge coloring of a general digraph D is a (not necessarily proper) coloring of its edges such that for each vertex v ∈ V ( D ) at least one color c satisfies the following conditions: if d D − ( v ) > 0 then c appears an odd number of times on the incoming edges at v ; and if d D + ( v ) > 0 then c appears an odd number of times on the outgoing edges at v . The minimum number of colors sufficient for a weak-odd edge coloring of D is the weak-odd chromatic index, denoted χ ′ wo ( D ) . It is known that χ ′ wo ( D ) ≤ 3 for every digraph D , and that the bound is sharp. In this article we show that the weak-odd chromatic index can be determined in polynomial time. Restricting to edge colorings of D with at most two colors, the minimum number of vertices v ∈ V ( D ) for which no color c satisfies the above conditions is the defect of D , denoted def( D ) . Surprisingly, it turns out that the problem of determining the defect of digraphs is (polynomially) equivalent to the problem of finding the matching number of simple graphs. Moreover, we characterize the classes of associated digraphs and tournaments in terms of the weak-odd chromatic index and the defect.
一般有向图D的弱奇边着色是其边的一种(不一定是固有的)着色，使得对于每个顶点v∈v (D)，至少有一种颜色c满足以下条件:如果D D−(v) > 0，则c在v处的入边上出现奇数次;如果dd + (v) > 0，则c在v处的出边出现奇数次。D的弱奇边着色所需的最小颜色数为弱奇色指数，记为χ ' wo (D)。已知对每一个有向图D， χ ' wo (D)≤3，且界是尖锐的。在本文中，我们证明了弱奇色指数可以在多项式时间内确定。在D的边着色最多有两种颜色的限制下，没有颜色c满足上述条件的顶点v∈v (D)的最小个数为D的缺陷，记为def(D)。令人惊讶的是，结果证明，确定有向图的缺陷的问题(多项式地)等同于寻找匹配数量的简单图的问题。此外，我们还根据弱奇色指数和缺陷对相关有向图和竞赛类进行了表征。
{"title":"Notes on weak-odd edge colorings of digraphs","authors":"César Hernández-Cruz, M. Petrusevski, R. Škrekovski","doi":"10.26493/1855-3974.1955.1cd","DOIUrl":"https://doi.org/10.26493/1855-3974.1955.1cd","url":null,"abstract":"A weak-odd edge coloring of a general digraph D is a (not necessarily proper) coloring of its edges such that for each vertex v ∈ V ( D ) at least one color c satisfies the following conditions: if d D − ( v ) > 0 then c appears an odd number of times on the incoming edges at v ; and if d D + ( v ) > 0 then c appears an odd number of times on the outgoing edges at v . The minimum number of colors sufficient for a weak-odd edge coloring of D is the weak-odd chromatic index, denoted χ ′ wo ( D ) . It is known that χ ′ wo ( D ) ≤ 3 for every digraph D , and that the bound is sharp. In this article we show that the weak-odd chromatic index can be determined in polynomial time. Restricting to edge colorings of D with at most two colors, the minimum number of vertices v ∈ V ( D ) for which no color c satisfies the above conditions is the defect of D , denoted def( D ) . Surprisingly, it turns out that the problem of determining the defect of digraphs is (polynomially) equivalent to the problem of finding the matching number of simple graphs. Moreover, we characterize the classes of associated digraphs and tournaments in terms of the weak-odd chromatic index and the defect.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"9 1","pages":"2"},"PeriodicalIF":0.0,"publicationDate":"2021-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89308176","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-22DOI: 10.26493/1855-3974.2522.EB3
Aleksandra Gorzkowska, Michael A. Henning, M. Pilsniak, Elżbieta Tumidajewicz
A set S of vertices in a graph G is a paired dominating set if every vertex of G is adjacent to a vertex in S and the subgraph induced by S contains a perfect matching (not necessarily as an induced subgraph). The paired domination number, γ pr ( G ) , of G is the minimum cardinality of a paired dominating set of G . A set of vertices whose removal from G produces a graph without isolated vertices is called a non-isolating set. The minimum cardinality of a non-isolating set of vertices whose removal decreases the paired domination number is the γ pr − -stability of G , denoted st γ pr − ( G ) . The paired domination stability of G is the minimum cardinality of a non-isolating set of vertices in G whose removal changes the paired domination number. We establish properties of paired domination stability in graphs. We prove that if G is a connected graph with γ pr ( G ) ≥ 4 , then st γ pr − ( G ) ≤ 2 Δ ( G ) where Δ ( G ) is the maximum degree in G , and we characterize the infinite family of trees that achieve equality in this upper bound.
{"title":"Paired domination stability in graphs","authors":"Aleksandra Gorzkowska, Michael A. Henning, M. Pilsniak, Elżbieta Tumidajewicz","doi":"10.26493/1855-3974.2522.EB3","DOIUrl":"https://doi.org/10.26493/1855-3974.2522.EB3","url":null,"abstract":"A set S of vertices in a graph G is a paired dominating set if every vertex of G is adjacent to a vertex in S and the subgraph induced by S contains a perfect matching (not necessarily as an induced subgraph). The paired domination number, γ pr ( G ) , of G is the minimum cardinality of a paired dominating set of G . A set of vertices whose removal from G produces a graph without isolated vertices is called a non-isolating set. The minimum cardinality of a non-isolating set of vertices whose removal decreases the paired domination number is the γ pr − -stability of G , denoted st γ pr − ( G ) . The paired domination stability of G is the minimum cardinality of a non-isolating set of vertices in G whose removal changes the paired domination number. We establish properties of paired domination stability in graphs. We prove that if G is a connected graph with γ pr ( G ) ≥ 4 , then st γ pr − ( G ) ≤ 2 Δ ( G ) where Δ ( G ) is the maximum degree in G , and we characterize the infinite family of trees that achieve equality in this upper bound.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"59 1","pages":"2"},"PeriodicalIF":0.0,"publicationDate":"2021-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75337873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-21DOI: 10.26493/1855-3974.2501.4C4
D. Crnković, Sanja Rukavina, Marina Simac
In this paper we study LDPC codes having cubic semisymmetric graphs as their Tanner graphs. We discuss the structure of the smallest absorbing sets of these LDPC codes. Further, we give an expression for the variance of the syndrome weight of the constructed codes, and present computational and simulation results.
{"title":"LDPC codes from cubic semisymmetric graphs","authors":"D. Crnković, Sanja Rukavina, Marina Simac","doi":"10.26493/1855-3974.2501.4C4","DOIUrl":"https://doi.org/10.26493/1855-3974.2501.4C4","url":null,"abstract":"In this paper we study LDPC codes having cubic semisymmetric graphs as their Tanner graphs. We discuss the structure of the smallest absorbing sets of these LDPC codes. Further, we give an expression for the variance of the syndrome weight of the constructed codes, and present computational and simulation results.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"19 1","pages":"2"},"PeriodicalIF":0.0,"publicationDate":"2021-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82870940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-12DOI: 10.26493/1855-3974.2332.749
A. Abdollahi, N. Zakeri
Let G be a graph on n vertices and consider the adjacency spectrum of G as the ordered n -tuple whose entries are eigenvalues of G written decreasingly. Let G and H be two non-isomorphic graphs on n vertices with spectra S and T , respectively. Define the distance between the spectra of G and H as the distance of S and T to a norm N of the n -dimensional vector space over real numbers. Define the cospectrality of G as the minimum of distances between the spectrum of G and spectra of all other non-isomorphic n vertices graphs to the norm N . In this paper we investigate copsectralities of the cocktail party graph and the complete tripartite graph with parts of the same size to the Euclidean or Manhattan norms.
{"title":"Cospectrality of multipartite graphs","authors":"A. Abdollahi, N. Zakeri","doi":"10.26493/1855-3974.2332.749","DOIUrl":"https://doi.org/10.26493/1855-3974.2332.749","url":null,"abstract":"Let G be a graph on n vertices and consider the adjacency spectrum of G as the ordered n -tuple whose entries are eigenvalues of G written decreasingly. Let G and H be two non-isomorphic graphs on n vertices with spectra S and T , respectively. Define the distance between the spectra of G and H as the distance of S and T to a norm N of the n -dimensional vector space over real numbers. Define the cospectrality of G as the minimum of distances between the spectrum of G and spectra of all other non-isomorphic n vertices graphs to the norm N . In this paper we investigate copsectralities of the cocktail party graph and the complete tripartite graph with parts of the same size to the Euclidean or Manhattan norms.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"98-100 1","pages":"1"},"PeriodicalIF":0.0,"publicationDate":"2021-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72552191","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-09DOI: 10.26493/1855-3974.2351.07B
Lowell Abrams, L. Lauderdale
The sum of distances between every pair of vertices in a graph G is called the Wiener index of G . This graph invariant was initially utilized to predict certain physico-chemical properties of organic compounds. However, the Wiener index of G does not account for any of its symmetries, which are also known to effect these physico-chemical properties. Graovac and Pisanski modified the Wiener index of G to measure the average distance each vertex is displaced under the elements of the symmetry group of G ; we call this the Graovac-Pisanski (GP) distance number of G . In this article, we prove that the set of all GP distance numbers of graphs with isomorphic symmetry groups is dense in a half-line. Moreover, for each finite group Γ and each rational number q within this half-line, we present a construction for a graph whose GP distance number is q and whose symmetry group is isomorphic to Γ . This construction results in graphs whose vertex orbits are not connected; we also consider an analogous construction which ensures that all vertex orbits are connected.
{"title":"Density results for Graovac-Pisanski's distance number","authors":"Lowell Abrams, L. Lauderdale","doi":"10.26493/1855-3974.2351.07B","DOIUrl":"https://doi.org/10.26493/1855-3974.2351.07B","url":null,"abstract":"The sum of distances between every pair of vertices in a graph G is called the Wiener index of G . This graph invariant was initially utilized to predict certain physico-chemical properties of organic compounds. However, the Wiener index of G does not account for any of its symmetries, which are also known to effect these physico-chemical properties. Graovac and Pisanski modified the Wiener index of G to measure the average distance each vertex is displaced under the elements of the symmetry group of G ; we call this the Graovac-Pisanski (GP) distance number of G . In this article, we prove that the set of all GP distance numbers of graphs with isomorphic symmetry groups is dense in a half-line. Moreover, for each finite group Γ and each rational number q within this half-line, we present a construction for a graph whose GP distance number is q and whose symmetry group is isomorphic to Γ . This construction results in graphs whose vertex orbits are not connected; we also consider an analogous construction which ensures that all vertex orbits are connected.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84427128","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-09DOI: 10.26493/1855-3974.2473.F2E
Z. Du, D. Dimitrov, C. Fonseca
We provide new families of divisibility and strong divisibility sequences based on some factorization properties of Chebyshev polynomials.
基于切比雪夫多项式的分解性质，给出了新的可整除性族和强可整除性序列。
{"title":"New strong divisibility sequences","authors":"Z. Du, D. Dimitrov, C. Fonseca","doi":"10.26493/1855-3974.2473.F2E","DOIUrl":"https://doi.org/10.26493/1855-3974.2473.F2E","url":null,"abstract":"We provide new families of divisibility and strong divisibility sequences based on some factorization properties of Chebyshev polynomials.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"31 1","pages":"1"},"PeriodicalIF":0.0,"publicationDate":"2021-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76626762","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-09DOI: 10.26493/1855-3974.2083.E80
E. Drgas-Burchardt, Agata Drzystek, E. Sidorowicz
Given a hypergraph ℋ and a function f : V (ℋ) → ℕ , we say that ℋ is f -choosable if there is a proper vertex coloring ϕ of ℋ such that ϕ ( v ) ∈ L ( v ) for all v ∈ V (ℋ) , where L : V (ℋ) → 2 ℕ is any assignment of f ( v ) colors to a vertex v . The sum choice number χ s c (ℋ) of ℋ is defined to be the minimum of ∑ v ∈ V (ℋ) f ( v ) over all functions f such that ℋ is f -choosable. A trivial upper bound on χ s c (ℋ) is | V (ℋ)| + |ℰ(ℋ)| . The class Γ s c of hypergraphs that achieve this bound is induced hereditary. We analyze some properties of hypergraphs in Γ s c as well as properties of hypergraphs in the class of forbidden hypergraphs for Γ s c . We characterize all θ -hypergraphs in Γ s c , which leads to the characterization of all θ -hypergraphs that are forbidden for Γ s c .
给定一个超图h和一个函数f: V (h)→n，如果存在一个合适的顶点着色φ，使得φ (V)∈L (V)对于所有V∈V (h)，其中L: V (h)→2 n是f (V)对顶点V的任意赋值，则我们说h是f -可选的。h的和选择数χ s c (h)定义为∑v∈v (h) f (v)在所有函数f上的最小值，使得h是f可选的。一个微不足道的上限χs c(ℋ)| V(ℋ)| + |ℰ(ℋ)|。该类Γ s c的超图达到这个界限是诱导遗传的。我们分析了Γ s c中超图的一些性质，以及Γ s c中禁忌超图类中的超图的性质。我们对Γ s c中的所有θ -超图进行了表征，从而得到了Γ s c中禁止的所有θ -超图的表征。
{"title":"Sum-list-colouring of θ-hypergraphs","authors":"E. Drgas-Burchardt, Agata Drzystek, E. Sidorowicz","doi":"10.26493/1855-3974.2083.E80","DOIUrl":"https://doi.org/10.26493/1855-3974.2083.E80","url":null,"abstract":"Given a hypergraph ℋ and a function f : V (ℋ) → ℕ , we say that ℋ is f -choosable if there is a proper vertex coloring ϕ of ℋ such that ϕ ( v ) ∈ L ( v ) for all v ∈ V (ℋ) , where L : V (ℋ) → 2 ℕ is any assignment of f ( v ) colors to a vertex v . The sum choice number χ s c (ℋ) of ℋ is defined to be the minimum of ∑ v ∈ V (ℋ) f ( v ) over all functions f such that ℋ is f -choosable. A trivial upper bound on χ s c (ℋ) is | V (ℋ)| + |ℰ(ℋ)| . The class Γ s c of hypergraphs that achieve this bound is induced hereditary. We analyze some properties of hypergraphs in Γ s c as well as properties of hypergraphs in the class of forbidden hypergraphs for Γ s c . We characterize all θ -hypergraphs in Γ s c , which leads to the characterization of all θ -hypergraphs that are forbidden for Γ s c .","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"350 1","pages":"1"},"PeriodicalIF":0.0,"publicationDate":"2021-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74828058","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Abreu, John Baptist Gauci, D. Labbate, F. Romaniello, J. P. Zerafa
A graph $G$ has the Perfect-Matching-Hamiltonian property (PMH-property) if for each one of its perfect matchings, there is another perfect matching of $G$ such that the union of the two perfect matchings yields a Hamiltonian cycle of $G$. The study of graphs that have the PMH-property, initiated in the 1970s by Las Vergnas and H"{a}ggkvist, combines three well-studied properties of graphs, namely matchings, Hamiltonicity and edge-colourings. In this work, we study these concepts for cubic graphs in an attempt to characterise those cubic graphs for which every perfect matching corresponds to one of the colours of a proper 3-edge-colouring of the graph. We discuss that this is equivalent to saying that such graphs are even-2-factorable (E2F), that is, all 2-factors of the graph contain only even cycles. The case for bipartite cubic graphs is trivial, since if $G$ is bipartite then it is E2F. Thus, we restrict our attention to non-bipartite cubic graphs. A sufficient, but not necessary, condition for a cubic graph to be E2F is that it has the PMH-property. The aim of this work is to introduce an infinite family of E2F non-bipartite cubic graphs on two parameters, which we coin papillon graphs, and determine the values of the respective parameters for which these graphs have the PMH-property or are just E2F. We also show that no two papillon graphs with different parameters are isomorphic.
{"title":"Perfect matchings, Hamiltonian cycles and edge-colourings in a class of cubic graphs","authors":"M. Abreu, John Baptist Gauci, D. Labbate, F. Romaniello, J. P. Zerafa","doi":"10.26493/1855-3974.23_3","DOIUrl":"https://doi.org/10.26493/1855-3974.23_3","url":null,"abstract":"A graph $G$ has the Perfect-Matching-Hamiltonian property (PMH-property) if for each one of its perfect matchings, there is another perfect matching of $G$ such that the union of the two perfect matchings yields a Hamiltonian cycle of $G$. The study of graphs that have the PMH-property, initiated in the 1970s by Las Vergnas and H\"{a}ggkvist, combines three well-studied properties of graphs, namely matchings, Hamiltonicity and edge-colourings. In this work, we study these concepts for cubic graphs in an attempt to characterise those cubic graphs for which every perfect matching corresponds to one of the colours of a proper 3-edge-colouring of the graph. We discuss that this is equivalent to saying that such graphs are even-2-factorable (E2F), that is, all 2-factors of the graph contain only even cycles. The case for bipartite cubic graphs is trivial, since if $G$ is bipartite then it is E2F. Thus, we restrict our attention to non-bipartite cubic graphs. A sufficient, but not necessary, condition for a cubic graph to be E2F is that it has the PMH-property. The aim of this work is to introduce an infinite family of E2F non-bipartite cubic graphs on two parameters, which we coin papillon graphs, and determine the values of the respective parameters for which these graphs have the PMH-property or are just E2F. We also show that no two papillon graphs with different parameters are isomorphic.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"70 12","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91406502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-13DOI: 10.26493/1855-3974.2129.AC1
Kan Hu, Young Soo Kwon, Jun-Yang Zhang
The auto-index of a skew morphism φ of a finite group A is the smallest positive integer h such that φ h is an automorphism of A . In this paper we develop a theory of auto-index of skew morphisms, and as an application we present a complete classification of skew morphisms of finite cyclic groups which are square roots of automorphisms.
{"title":"Classification of skew morphisms of cyclic groups which are square roots of automorphisms","authors":"Kan Hu, Young Soo Kwon, Jun-Yang Zhang","doi":"10.26493/1855-3974.2129.AC1","DOIUrl":"https://doi.org/10.26493/1855-3974.2129.AC1","url":null,"abstract":"The auto-index of a skew morphism φ of a finite group A is the smallest positive integer h such that φ h is an automorphism of A . In this paper we develop a theory of auto-index of skew morphisms, and as an application we present a complete classification of skew morphisms of finite cyclic groups which are square roots of automorphisms.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84183410","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-03DOI: 10.26493/1855-3974.2659.be1
M. A. Ollis, A. Pasotti, M. Pellegrini, John R. Schmitt
Label the vertices of the complete graph Kv with the integers {0, 1, . . . , v − 1} and define the length of the edge between x and y to be min(|x−y|, v−|x−y|). Let L be a multiset of size v − 1 with underlying set contained in {1, . . . , bv/2c}. The Buratti-Horak-Rosa Conjecture is that there is a Hamiltonian path in Kv whose edge lengths are exactly L if and only if for any divisor d of v the number of multiples of d appearing in L is at most v − d. We introduce “growable realizations,” which enable us to prove many new instances of the conjecture and to reprove known results in a simpler way. As examples of the new method, we give a complete solution when the underlying set is contained in {1, 4, 5} or in {1, 2, 3, 4} and a partial result when the underlying set has the form {1, x, 2x}. We believe that for any set U of positive integers there is a finite set of growable realizations that implies the truth of the Buratti-Horak-Rosa Conjecture for all but finitely many multisets with underlying set U . MSC: 05C38, 05C78.
{"title":"Growable realizations: a powerful approach to the Buratti-Horak-Rosa Conjecture","authors":"M. A. Ollis, A. Pasotti, M. Pellegrini, John R. Schmitt","doi":"10.26493/1855-3974.2659.be1","DOIUrl":"https://doi.org/10.26493/1855-3974.2659.be1","url":null,"abstract":"Label the vertices of the complete graph Kv with the integers {0, 1, . . . , v − 1} and define the length of the edge between x and y to be min(|x−y|, v−|x−y|). Let L be a multiset of size v − 1 with underlying set contained in {1, . . . , bv/2c}. The Buratti-Horak-Rosa Conjecture is that there is a Hamiltonian path in Kv whose edge lengths are exactly L if and only if for any divisor d of v the number of multiples of d appearing in L is at most v − d. We introduce “growable realizations,” which enable us to prove many new instances of the conjecture and to reprove known results in a simpler way. As examples of the new method, we give a complete solution when the underlying set is contained in {1, 4, 5} or in {1, 2, 3, 4} and a partial result when the underlying set has the form {1, x, 2x}. We believe that for any set U of positive integers there is a finite set of growable realizations that implies the truth of the Buratti-Horak-Rosa Conjecture for all but finitely many multisets with underlying set U . MSC: 05C38, 05C78.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"41 1","pages":"4"},"PeriodicalIF":0.0,"publicationDate":"2021-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82398874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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