Let π = (f1, . . . , fm; g1, . . . , gn), where f1, . . . , fm and g1, . . . , gn are two non-increasing sequences of nonnegative integers. The pair π = (f1, . . . , fm; g1, . . . , gn) is said to be a bigraphic pair if there is a simple bipartite graph G = (X ∪ Y,E) such that f1, . . . , fm and g1, . . . , gn are the degrees of the vertices in X and Y , respectively. In this case, G is referred to as a realization of π. We say that π is a potentially Ks,t-bigraphic pair if some realization of π contains Ks,t (with s vertices in the part of size m and t in the part of size n). Ferrara et al. [Potentially H-bigraphic sequences, Discuss. Math. Graph Theory 29 (2009) 583–596] defined σ(Ks,t,m, n) to be the minimum integer k such that every bigraphic pair π = (f1, . . . , fm; g1, . . . , gn) with σ(π) = f1+· · ·+fm ≥ k is potentiallyKs,t-bigraphic. They determined σ(Ks,t,m, n) for n ≥ m ≥ 9st. In this paper, we first give a procedure and two sufficient conditions to determine if π is a potentially Ks,t-bigraphic pair. Then, we determine σ(Ks,t,m, n) for n ≥ m ≥ s and n ≥ (s+ 1)t− (2s− 1)t+ s− 1. This provides a solution to a problem due to Ferrara et al.
{"title":"About an extremal problem of bigraphic pairs with a realization containing Ks, t","authors":"Jianhua Yin, Bing Wang","doi":"10.7151/dmgt.2375","DOIUrl":"https://doi.org/10.7151/dmgt.2375","url":null,"abstract":"Let π = (f1, . . . , fm; g1, . . . , gn), where f1, . . . , fm and g1, . . . , gn are two non-increasing sequences of nonnegative integers. The pair π = (f1, . . . , fm; g1, . . . , gn) is said to be a bigraphic pair if there is a simple bipartite graph G = (X ∪ Y,E) such that f1, . . . , fm and g1, . . . , gn are the degrees of the vertices in X and Y , respectively. In this case, G is referred to as a realization of π. We say that π is a potentially Ks,t-bigraphic pair if some realization of π contains Ks,t (with s vertices in the part of size m and t in the part of size n). Ferrara et al. [Potentially H-bigraphic sequences, Discuss. Math. Graph Theory 29 (2009) 583–596] defined σ(Ks,t,m, n) to be the minimum integer k such that every bigraphic pair π = (f1, . . . , fm; g1, . . . , gn) with σ(π) = f1+· · ·+fm ≥ k is potentiallyKs,t-bigraphic. They determined σ(Ks,t,m, n) for n ≥ m ≥ 9st. In this paper, we first give a procedure and two sufficient conditions to determine if π is a potentially Ks,t-bigraphic pair. Then, we determine σ(Ks,t,m, n) for n ≥ m ≥ s and n ≥ (s+ 1)t− (2s− 1)t+ s− 1. This provides a solution to a problem due to Ferrara et al.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90368554","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}
David A. Kalarkop, I. Sahul Hamid, M. Chellali, R. Rangarajan
{"title":"Global dominated coloring of graphs","authors":"David A. Kalarkop, I. Sahul Hamid, M. Chellali, R. Rangarajan","doi":"10.7151/dmgt.2498","DOIUrl":"https://doi.org/10.7151/dmgt.2498","url":null,"abstract":"","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71129955","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 main eigenvalues of a graph G are those eigenvalues of the (0, 1)adjacency matrix A with a corresponding eigenspace not orthogonal to j = (1 | 1 | · · · | 1). The principal main eigenvector associated with a main eigenvalue is the orthogonal projection of the corresponding eigenspace onto j. The main eigenspace of a graph is generated by all the principal main eigenvectors and is the same as the image of the walk matrix. We explore a new concept to see to what extent the main eigenspace determines the entries of the walk matrix of a graph. The CDC of a graph G is the direct product G ×K2. We establish a hierarchy of inclusions connecting classes of graphs in view of their CDC, walk matrix, main eigenvalues and main eigenspaces. We provide a new proof that graphs with the same CDC are characterized as TF-isomorphic graphs. A complete list of TF-isomorphic graphs on at most 8 vertices and their common CDC is also given.
{"title":"The walks and CDC of graphs with the same main eigenspace","authors":"Irene Sciriha, Luke Collins","doi":"10.7151/dmgt.2386","DOIUrl":"https://doi.org/10.7151/dmgt.2386","url":null,"abstract":"The main eigenvalues of a graph G are those eigenvalues of the (0, 1)adjacency matrix A with a corresponding eigenspace not orthogonal to j = (1 | 1 | · · · | 1). The principal main eigenvector associated with a main eigenvalue is the orthogonal projection of the corresponding eigenspace onto j. The main eigenspace of a graph is generated by all the principal main eigenvectors and is the same as the image of the walk matrix. We explore a new concept to see to what extent the main eigenspace determines the entries of the walk matrix of a graph. The CDC of a graph G is the direct product G ×K2. We establish a hierarchy of inclusions connecting classes of graphs in view of their CDC, walk matrix, main eigenvalues and main eigenspaces. We provide a new proof that graphs with the same CDC are characterized as TF-isomorphic graphs. A complete list of TF-isomorphic graphs on at most 8 vertices and their common CDC is also given.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73706417","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}
Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $Ssubseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $gamma(G)$, is the domination number of $G$. Two vertices are neighbors if they are adjacent. A super dominating set is a dominating set $S$ with the additional property that every vertex in $V setminus S$ has a neighbor in $S$ that is adjacent to no other vertex in $V setminus S$. Moreover if every vertex in $V setminus S$ has degree at least~$2$, then $S$ is an end super dominating set. The end super domination number is the minimum cardinality of an end super dominating set. We give applications of end super dominating sets as main servers and temporary servers of networks. We determine the exact value of the end super domination number for specific classes of graphs, and we count the number of end super dominating sets in these graphs. Tight upper bounds on the end super domination number are established, where the graph is modified by vertex (edge) removal and contraction.
{"title":"End super dominating sets in graphs","authors":"Saieed Akbari, Nima Ghanbari, Michael A. Henning","doi":"10.7151/dmgt.2519","DOIUrl":"https://doi.org/10.7151/dmgt.2519","url":null,"abstract":"Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $Ssubseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $gamma(G)$, is the domination number of $G$. Two vertices are neighbors if they are adjacent. A super dominating set is a dominating set $S$ with the additional property that every vertex in $V setminus S$ has a neighbor in $S$ that is adjacent to no other vertex in $V setminus S$. Moreover if every vertex in $V setminus S$ has degree at least~$2$, then $S$ is an end super dominating set. The end super domination number is the minimum cardinality of an end super dominating set. We give applications of end super dominating sets as main servers and temporary servers of networks. We determine the exact value of the end super domination number for specific classes of graphs, and we count the number of end super dominating sets in these graphs. Tight upper bounds on the end super domination number are established, where the graph is modified by vertex (edge) removal and contraction.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135909489","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 lexicographic product $G[H]$ of two graphs $G$ and $H$ is obtained from $G$ by replacing each vertex with a copy of $H$ and adding all edges between any pair of copies corresponding to adjacent vertices of $G$. We generalize the lexicographic product such that we replace each vertex of $G$ with arbitrary graph on the same number of vertices. We present sufficient and necessary conditions for traceability, hamiltonicity and hamiltonian connectivity of $G[H]$ if $G$ is a path.
{"title":"Hamiltonian properties in generalized lexicographic products","authors":"Jan Ekstein, Jakub Teska","doi":"10.7151/dmgt.2527","DOIUrl":"https://doi.org/10.7151/dmgt.2527","url":null,"abstract":"The lexicographic product $G[H]$ of two graphs $G$ and $H$ is obtained from $G$ by replacing each vertex with a copy of $H$ and adding all edges between any pair of copies corresponding to adjacent vertices of $G$. We generalize the lexicographic product such that we replace each vertex of $G$ with arbitrary graph on the same number of vertices. We present sufficient and necessary conditions for traceability, hamiltonicity and hamiltonian connectivity of $G[H]$ if $G$ is a path.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135784462","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}
In this paper the behavior of the game domination number γg(G) and the Staller start game domination number γ′ g(G) by the contraction of an edge and the subdivision of an edge are investigated. Here we prove that contracting an edge can decrease γg(G) and γ ′ g(G) by at most two, whereas subdividing an edge can increase these parameters by at most two. In the case of no-minus graphs it is proved that subdividing an edge can increase both these parameters by at most one but on the other hand contracting an edge can decrease these by two. All possible values of these parameters are also analysed here.
本文研究了由边的收缩和边的细分得到的对策支配数γ G (G)和Staller起始对策支配数γ ' G (G)的行为。在这里，我们证明了收缩一条边可以使γ G (G)和γ ' G (G)减少至多两个，而细分一条边则可以使这些参数增加至多两个。在无负图的情况下，证明了细分一条边可以使这两个参数最多增加一个，而收缩一条边则可以使这两个参数减少两个。这里还分析了这些参数的所有可能值。
{"title":"Domination game: Effect of edge contraction and edge subdivision","authors":"Tijo James, A. Vijayakumar","doi":"10.7151/dmgt.2378","DOIUrl":"https://doi.org/10.7151/dmgt.2378","url":null,"abstract":"In this paper the behavior of the game domination number γg(G) and the Staller start game domination number γ′ g(G) by the contraction of an edge and the subdivision of an edge are investigated. Here we prove that contracting an edge can decrease γg(G) and γ ′ g(G) by at most two, whereas subdividing an edge can increase these parameters by at most two. In the case of no-minus graphs it is proved that subdividing an edge can increase both these parameters by at most one but on the other hand contracting an edge can decrease these by two. All possible values of these parameters are also analysed here.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75899893","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}
{"title":"Bipartite Ramsey number pairs involving cycles","authors":"Ernst J. Joubert, Johannes Hattingh","doi":"10.7151/dmgt.2526","DOIUrl":"https://doi.org/10.7151/dmgt.2526","url":null,"abstract":"","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135703797","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 an isolate-free graph G = ( V ( G ) , E ( G )), a set S ⊆ V ( G ) is called a semitotal forcing set of G if it is a forcing set (or a zero forcing set) of G and every vertex in S is within distance 2 of another vertex of S . The semitotal forcing number F t 2 ( G ) is the minimum cardinality of a semitotal forcing set in G . In this paper, we prove that if G (cid:54) = K 4 is a connected claw-free cubic graph of order n , then F t 2 ( G ) ≤ 38 n + 1. The graphs achieving equality in this bound are characterized, an infinite set of graphs.
对于无隔离图G = (V (G)， E (G))，如果集合S是G的一个强迫集(或零强迫集)，且S中的每个顶点与S的另一个顶点的距离在2以内，则称其为G的半强迫集。半衰期强迫数f2 (G)是G中半衰期强迫集的最小基数。本文证明了如果G (cid:54) = k4是一个n阶的连通无爪三次图，则F t 2 (G)≤38n + 1。在这个界内达到相等的图被表示为图的无限集。
{"title":"Semitotal forcing in claw-free cubic graphs","authors":"Yijuan Liang, Jie Chen, Shou-Jun Xu","doi":"10.7151/dmgt.2501","DOIUrl":"https://doi.org/10.7151/dmgt.2501","url":null,"abstract":"For an isolate-free graph G = ( V ( G ) , E ( G )), a set S ⊆ V ( G ) is called a semitotal forcing set of G if it is a forcing set (or a zero forcing set) of G and every vertex in S is within distance 2 of another vertex of S . The semitotal forcing number F t 2 ( G ) is the minimum cardinality of a semitotal forcing set in G . In this paper, we prove that if G (cid:54) = K 4 is a connected claw-free cubic graph of order n , then F t 2 ( G ) ≤ 38 n + 1. The graphs achieving equality in this bound are characterized, an infinite set of graphs.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71129611","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}
{"title":"Spectral bounds for the zero forcing number of a graph","authors":"Hongzhang Chen, Jianxi Li, Shou-Jun Xu","doi":"10.7151/dmgt.2482","DOIUrl":"https://doi.org/10.7151/dmgt.2482","url":null,"abstract":"","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71129810","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 revisit the Burr–Erd˝os–Lov´asz conjecture on chromatic Ramsey numbers. We show that it admits a proof based on the Lov´asz ϑ parame- ter in addition to the proof of Xuding Zhu based on the fractional chromatic number. However, there are no proofs based on topological lower bounds on chromatic numbers, because the chromatic Ramsey numbers of generalised Mycielski graphs are too large. We show that the 4-chromatic generalised Mycielski graphs other than K 4 all have chromatic Ramsey number 14, and that the n -chromatic generalised Mycielski graphs all have chromatic Ramsey number at least 2 n/ 4 .
{"title":"Chromatic Ramsey numbers of generalised Mycielski graphs","authors":"Claude Tardif","doi":"10.7151/dmgt.2499","DOIUrl":"https://doi.org/10.7151/dmgt.2499","url":null,"abstract":". We revisit the Burr–Erd˝os–Lov´asz conjecture on chromatic Ramsey numbers. We show that it admits a proof based on the Lov´asz ϑ parame- ter in addition to the proof of Xuding Zhu based on the fractional chromatic number. However, there are no proofs based on topological lower bounds on chromatic numbers, because the chromatic Ramsey numbers of generalised Mycielski graphs are too large. We show that the 4-chromatic generalised Mycielski graphs other than K 4 all have chromatic Ramsey number 14, and that the n -chromatic generalised Mycielski graphs all have chromatic Ramsey number at least 2 n/ 4 .","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71129970","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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