Pub Date : 2024-07-25DOI: 10.26493/1855-3974.3015.f4a
Diane Castonguay, Erika M. M. Coelho, Hebert Coelho, J. Nascimento, Uéverton S. Souza
{"title":"Perfect matching cuts partitioning a graph into complementary subgraphs","authors":"Diane Castonguay, Erika M. M. Coelho, Hebert Coelho, J. Nascimento, Uéverton S. Souza","doi":"10.26493/1855-3974.3015.f4a","DOIUrl":"https://doi.org/10.26493/1855-3974.3015.f4a","url":null,"abstract":"","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141805323","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 : 2024-07-15DOI: 10.26493/1855-3974.3227.fd5
Zoran Stanić
{"title":"Spectra of signed graphs and related oriented graphs","authors":"Zoran Stanić","doi":"10.26493/1855-3974.3227.fd5","DOIUrl":"https://doi.org/10.26493/1855-3974.3227.fd5","url":null,"abstract":"","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141646735","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 : 2023-12-15DOI: 10.26493/1855-3974.3229.8b1
M. Conder, Isabelle Steinmann
{"title":"Answers to questions about medial layer graphs of self-dual regular and chiral polytopes","authors":"M. Conder, Isabelle Steinmann","doi":"10.26493/1855-3974.3229.8b1","DOIUrl":"https://doi.org/10.26493/1855-3974.3229.8b1","url":null,"abstract":"","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138998433","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 : 2023-11-08DOI: 10.26493/1855-3974.2992.a6d
Yanan Hu, Xingzhi Zhan
The three subgraphs of a connected graph induced by the center, annulus and periphery are called its metric subgraphs. The main results are as follows. (1) There exists a graph of order n whose metric subgraphs are all paths if and only if n ≥ 13 and the smallest size of such a graph of order 13 is 22; (2) there exists a graph of order n whose metric subgraphs are all cycles if and only if n ≥ 15, and there are exactly three such graphs of order 15; (3) for every integer k ≥ 3, we determine the possible orders for the existence of a graph whose metric subgraphs are all connected k-regular graphs; (4) there exists a graph of order n whose metric subgraphs are connected and pairwise isomorphic if and only if n ≥ 24 and n is divisible by 3. An unsolved problem is posed.
{"title":"On the metric subgraphs of a graph","authors":"Yanan Hu, Xingzhi Zhan","doi":"10.26493/1855-3974.2992.a6d","DOIUrl":"https://doi.org/10.26493/1855-3974.2992.a6d","url":null,"abstract":"The three subgraphs of a connected graph induced by the center, annulus and periphery are called its metric subgraphs. The main results are as follows. (1) There exists a graph of order n whose metric subgraphs are all paths if and only if n ≥ 13 and the smallest size of such a graph of order 13 is 22; (2) there exists a graph of order n whose metric subgraphs are all cycles if and only if n ≥ 15, and there are exactly three such graphs of order 15; (3) for every integer k ≥ 3, we determine the possible orders for the existence of a graph whose metric subgraphs are all connected k-regular graphs; (4) there exists a graph of order n whose metric subgraphs are connected and pairwise isomorphic if and only if n ≥ 24 and n is divisible by 3. An unsolved problem is posed.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135342368","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 : 2023-11-06DOI: 10.26493/1855-3974.2408.42e
Robert Jajcay, Jozef Širáň, Yan Wang
Cayley maps are embeddings of Cayley graphs in orientable surfaces which possess a group of orientation preserving automorphisms acting regularly on the vertices. We generalize the concept of a Cayley map by considering embeddings of Cayley graphs in both orientable and non-orientable surfaces and by requiring a group of automorphisms acting regularly on vertices that does not have to consist entirely of orientation preserving automorphisms. This leads to new families of maps in both the orientable and non-orientable cases. Since the Petrie dual operator preserves the property of being a generalized Cayley map, throughout the paper we consider the action of this operator on our maps.
{"title":"Generalized Cayley maps and their Petrie duals","authors":"Robert Jajcay, Jozef Širáň, Yan Wang","doi":"10.26493/1855-3974.2408.42e","DOIUrl":"https://doi.org/10.26493/1855-3974.2408.42e","url":null,"abstract":"Cayley maps are embeddings of Cayley graphs in orientable surfaces which possess a group of orientation preserving automorphisms acting regularly on the vertices. We generalize the concept of a Cayley map by considering embeddings of Cayley graphs in both orientable and non-orientable surfaces and by requiring a group of automorphisms acting regularly on vertices that does not have to consist entirely of orientation preserving automorphisms. This leads to new families of maps in both the orientable and non-orientable cases. Since the Petrie dual operator preserves the property of being a generalized Cayley map, throughout the paper we consider the action of this operator on our maps.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135633995","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 : 2023-10-26DOI: 10.26493/1855-3974.2910.5b3
Simone Costa, Anita Pasotti
This article aims to provide exponential lower bounds on the number of non-isomorphic k-gonal face-2-colourable embeddings (sometimes called, with abuse of notation, biembeddings) of the complete multipartite graph into orientable surfaces. For this purpose, we use the concept, introduced by Archdeacon in 2015, of Heffer array and its relations with graph embeddings. In particular we show that, under certain hypotheses, from a single Heffter array, we can obtain an exponential number of distinct graph embeddings. Exploiting this idea starting from the arrays constructed by Cavenagh, Donovan and Yazıcı in 2020, we obtain that, for infinitely many values of k and v, there are at least kk/2 + o(k) ⋅ 2v ⋅ H(1/4)/(2k)^2 + o(v) non-isomorphic k-gonal face-2-colourable embeddings of Kv, where H(⋅) is the binary entropy. Moreover about the embeddings of Kv/t × t, for t ∈ {1, 2, k}, we provide a construction of 2v ⋅ H(1/4)/2k(k−1) + o(v,k) non-isomorphic k-gonal face-2-colourable embeddings whenever k is odd and v belongs to a wide infinite family of values.
{"title":"On the number of non-isomorphic (simple) k-gonal biembeddings of complete multipartite graphs","authors":"Simone Costa, Anita Pasotti","doi":"10.26493/1855-3974.2910.5b3","DOIUrl":"https://doi.org/10.26493/1855-3974.2910.5b3","url":null,"abstract":"This article aims to provide exponential lower bounds on the number of non-isomorphic k-gonal face-2-colourable embeddings (sometimes called, with abuse of notation, biembeddings) of the complete multipartite graph into orientable surfaces. For this purpose, we use the concept, introduced by Archdeacon in 2015, of Heffer array and its relations with graph embeddings. In particular we show that, under certain hypotheses, from a single Heffter array, we can obtain an exponential number of distinct graph embeddings. Exploiting this idea starting from the arrays constructed by Cavenagh, Donovan and Yazıcı in 2020, we obtain that, for infinitely many values of k and v, there are at least kk/2 + o(k) ⋅ 2v ⋅ H(1/4)/(2k)^2 + o(v) non-isomorphic k-gonal face-2-colourable embeddings of Kv, where H(⋅) is the binary entropy. Moreover about the embeddings of Kv/t × t, for t ∈ {1, 2, k}, we provide a construction of 2v ⋅ H(1/4)/2k(k−1) + o(v,k) non-isomorphic k-gonal face-2-colourable embeddings whenever k is odd and v belongs to a wide infinite family of values.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136377025","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 : 2023-10-23DOI: 10.26493/1855-3974.3009.6df
Ivan Damnjanović
A circulant nut graph is a non-trivial simple graph such that its adjacency matrix is a circulant matrix whose null space is spanned by a single vector without zero elements. Regarding these graphs, the order–degree existence problem can be thought of as the mathematical problem of determining all the possible pairs (n, d) for which there exists a d-regular circulant nut graph of order n. This problem was initiated by Bašić et al. and the first major results were obtained by Damnjanović and Stevanović, who proved that for each odd t ≥ 3 such that t ≢10 1 and t ≢18 15, there exists a 4t-regular circulant nut graph of order n for each even n ≥ 4t + 4. Afterwards, Damnjanović improved these results by showing that there necessarily exists a 4t-regular circulant nut graph of order n whenever t is odd, n is even, and n ≥ 4t + 4 holds, or whenever t is even, n is such that n ≡4 2, and n ≥ 4t + 6 holds. In this paper, we extend the aforementioned results by completely resolving the circulant nut graph order–degree existence problem. In other words, we fully determine all the possible pairs (n, d) for which there exists a d-regular circulant nut graph of order n.
{"title":"Complete resolution of the circulant nut graph order–degree existence problem","authors":"Ivan Damnjanović","doi":"10.26493/1855-3974.3009.6df","DOIUrl":"https://doi.org/10.26493/1855-3974.3009.6df","url":null,"abstract":"A circulant nut graph is a non-trivial simple graph such that its adjacency matrix is a circulant matrix whose null space is spanned by a single vector without zero elements. Regarding these graphs, the order–degree existence problem can be thought of as the mathematical problem of determining all the possible pairs (n, d) for which there exists a d-regular circulant nut graph of order n. This problem was initiated by Bašić et al. and the first major results were obtained by Damnjanović and Stevanović, who proved that for each odd t ≥ 3 such that t ≢10 1 and t ≢18 15, there exists a 4t-regular circulant nut graph of order n for each even n ≥ 4t + 4. Afterwards, Damnjanović improved these results by showing that there necessarily exists a 4t-regular circulant nut graph of order n whenever t is odd, n is even, and n ≥ 4t + 4 holds, or whenever t is even, n is such that n ≡4 2, and n ≥ 4t + 6 holds. In this paper, we extend the aforementioned results by completely resolving the circulant nut graph order–degree existence problem. In other words, we fully determine all the possible pairs (n, d) for which there exists a d-regular circulant nut graph of order n.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135406284","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 : 2023-10-03DOI: 10.26493/1855-3974.3109.e4b
Yair Caro, Balázs Patkós, Zsolt Tuza
The connected Turán number is a variant of the much studied Turán number, ex(n,F), the largest number of edges that an n-vertex F-free graph may contain. We start a systematic study of the connected Turán number exc(n,F), the largest number of edges that an n-vertex connected F-free graph may contain. We focus on the case where the forbidden graph is a tree. Prior to our work, exc(n,T) was determined only for the case T is a star or a path. Our main contribution is the determination of the exact value of exc(n,T) for small trees, in particular for all trees with at most six vertices, as well as some trees on seven vertices and several infinite families of trees. We also collect several lower-bound constructions of connected T-free graphs based on different graph parameters. The celebrated conjecture of Erdős and Sós states that for any tree T, we have ex(n,T) ≤ (|T|−2)n/2. We address the problem how much smaller exc(n,T) can be, what is the smallest possible ratio of exc(n,T) and (|T|−2)n/2 as |T| grows.
{"title":"Connected Turán number of trees","authors":"Yair Caro, Balázs Patkós, Zsolt Tuza","doi":"10.26493/1855-3974.3109.e4b","DOIUrl":"https://doi.org/10.26493/1855-3974.3109.e4b","url":null,"abstract":"The connected Turán number is a variant of the much studied Turán number, ex(n,F), the largest number of edges that an n-vertex F-free graph may contain. We start a systematic study of the connected Turán number exc(n,F), the largest number of edges that an n-vertex connected F-free graph may contain. We focus on the case where the forbidden graph is a tree. Prior to our work, exc(n,T) was determined only for the case T is a star or a path. Our main contribution is the determination of the exact value of exc(n,T) for small trees, in particular for all trees with at most six vertices, as well as some trees on seven vertices and several infinite families of trees. We also collect several lower-bound constructions of connected T-free graphs based on different graph parameters. The celebrated conjecture of Erdős and Sós states that for any tree T, we have ex(n,T) ≤ (|T|−2)n/2. We address the problem how much smaller exc(n,T) can be, what is the smallest possible ratio of exc(n,T) and (|T|−2)n/2 as |T| grows.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135738955","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 : 2023-09-27DOI: 10.26493/1855-3974.2619.06c
Javad Bagherian, Hanieh Memarzadeh
In this paper, we first introduce a new product of finite graphs as a generalization of the X-join of graphs. We then give the necessary and sufficient conditions under which a graph has the generalized X-join structure. As a main result, we compute the full automorphism groups of the family of graphs that have the generalized X-join structure.
{"title":"Generalized X-join of graphs and their automorphisms","authors":"Javad Bagherian, Hanieh Memarzadeh","doi":"10.26493/1855-3974.2619.06c","DOIUrl":"https://doi.org/10.26493/1855-3974.2619.06c","url":null,"abstract":"In this paper, we first introduce a new product of finite graphs as a generalization of the X-join of graphs. We then give the necessary and sufficient conditions under which a graph has the generalized X-join structure. As a main result, we compute the full automorphism groups of the family of graphs that have the generalized X-join structure.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135476074","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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