{"title":"All tight descriptions of faces in plane triangulations with minimum degree 4","authors":"A. Ivanova, O. Borodin","doi":"10.7151/dmgt.2488","DOIUrl":"https://doi.org/10.7151/dmgt.2488","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":"71130002","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 F be a family of graphs. The Turán number of F , denoted by ex(n,F), is the maximum number of edges in a graph with n vertices which does not contain any subgraph isomorphic to some graph in F . A star forest is a forest whose connected components are all stars and isolated vertices. Motivated by the results of Wang, Yang and Ning about the spanning Turán number of linear forests [J. Wang and W. Yang, The Turán number for spanning linear forests, Discrete Appl. Math. 254 (2019) 291–294; B. Ning and J. Wang, The formula for Turán number of spanning linear forests, Discrete Math. 343 (2020) 111924]. In this paper, let Sn,k be the set of all star forests with n vertices and k edges. We prove that when 1 ≤ k ≤ n− 1, ex(n,Sn,k) = ⌊ k−1 2 ⌋ .
{"title":"The Turán number of spanning star forests","authors":"Lin-Peng Zhang, Ligong Wang, Jiale Zhou","doi":"10.7151/dmgt.2368","DOIUrl":"https://doi.org/10.7151/dmgt.2368","url":null,"abstract":"Let F be a family of graphs. The Turán number of F , denoted by ex(n,F), is the maximum number of edges in a graph with n vertices which does not contain any subgraph isomorphic to some graph in F . A star forest is a forest whose connected components are all stars and isolated vertices. Motivated by the results of Wang, Yang and Ning about the spanning Turán number of linear forests [J. Wang and W. Yang, The Turán number for spanning linear forests, Discrete Appl. Math. 254 (2019) 291–294; B. Ning and J. Wang, The formula for Turán number of spanning linear forests, Discrete Math. 343 (2020) 111924]. In this paper, let Sn,k be the set of all star forests with n vertices and k edges. We prove that when 1 ≤ k ≤ n− 1, ex(n,Sn,k) = ⌊ k−1 2 ⌋ .","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":"78930417","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}
C. Bornstein, G. Morgenstern, T. Santos, U. Souza, J. Szwarcfiter
{"title":"Helly and strong Helly numbers of Bk-EPG and Bk-VPG graphs","authors":"C. Bornstein, G. Morgenstern, T. Santos, U. Souza, J. Szwarcfiter","doi":"10.7151/dmgt.2427","DOIUrl":"https://doi.org/10.7151/dmgt.2427","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":"88547678","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. Cabrera Martínez, José L. Sánchez, J. M. Sigarreta
Abstract Let G be a graph with no isolated vertex. A set D ⊆ V (G) is a total dominating set of G if every vertex of G is adjacent to at least one vertex in D. The total domination number of G, denoted by γt (G), is the minimum cardinality among all total dominating sets of G. In this paper we study the total domination number of total graphs T(G) of simple graphs G. In particular, we give some relationships that exist between γt(T(G)) and other domination parameters of G and of some well-known graph operators on G. Finally, we provide closed formulas on γt (T(G)) for some well-known families of graphs G.
{"title":"On the Total Domination Number of Total Graphs","authors":"A. Cabrera Martínez, José L. Sánchez, J. M. Sigarreta","doi":"10.7151/dmgt.2478","DOIUrl":"https://doi.org/10.7151/dmgt.2478","url":null,"abstract":"Abstract Let G be a graph with no isolated vertex. A set D ⊆ V (G) is a total dominating set of G if every vertex of G is adjacent to at least one vertex in D. The total domination number of G, denoted by γt (G), is the minimum cardinality among all total dominating sets of G. In this paper we study the total domination number of total graphs T(G) of simple graphs G. In particular, we give some relationships that exist between γt(T(G)) and other domination parameters of G and of some well-known graph operators on G. Finally, we provide closed formulas on γt (T(G)) for some well-known families of graphs G.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45519491","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}
Abstract In a (G1, G2) coloring of a graph G, every edge of G is in G1 or G2. For two bipartite graphs H1 and H2, the bipartite Ramsey number BR(H1, H2) is the least integer b ≥ 1, such that for every (G1, G2) coloring of the complete bipartite graph Kb,b, results in either H1 ⊆ G1 or H2 ⊆ G2. As another view, for bipartite graphs H1 and H2 and a positive integer m, the m-bipartite Ramsey number BRm(H1, H2) of H1 and H2 is the least integer n (n ≥ m) such that every subgraph G of Km,n results in H1 ⊆ G or H2 ⊆ Ḡ. The size of m-bipartite Ramsey number BRm(K2,2, K2,2), the size of m-bipartite Ramsey number BRm(K2,2, K3,3) and the size of m-bipartite Ramsey number BRm(K3,3, K3,3) have been computed in several articles up to now. In this paper we determine the exact value of BRm(K2,2, K4,4) for each m ≥ 2.
{"title":"The m-Bipartite Ramsey Number BRm(H1, H2)","authors":"Yaser Rowshan","doi":"10.7151/dmgt.2477","DOIUrl":"https://doi.org/10.7151/dmgt.2477","url":null,"abstract":"Abstract In a (G1, G2) coloring of a graph G, every edge of G is in G1 or G2. For two bipartite graphs H1 and H2, the bipartite Ramsey number BR(H1, H2) is the least integer b ≥ 1, such that for every (G1, G2) coloring of the complete bipartite graph Kb,b, results in either H1 ⊆ G1 or H2 ⊆ G2. As another view, for bipartite graphs H1 and H2 and a positive integer m, the m-bipartite Ramsey number BRm(H1, H2) of H1 and H2 is the least integer n (n ≥ m) such that every subgraph G of Km,n results in H1 ⊆ G or H2 ⊆ Ḡ. The size of m-bipartite Ramsey number BRm(K2,2, K2,2), the size of m-bipartite Ramsey number BRm(K2,2, K3,3) and the size of m-bipartite Ramsey number BRm(K3,3, K3,3) have been computed in several articles up to now. In this paper we determine the exact value of BRm(K2,2, K4,4) for each m ≥ 2.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41575632","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}
If $G$ is a graph and $Xsubseteq V(G)$, then $X$ is a total mutual-visibility set if every pair of vertices $x$ and $y$ of $G$ admits a shortest $x,y$-path $P$ with $V(P) cap X subseteq {x,y}$. The cardinality of a largest total mutual-visibility set of $G$ is the total mutual-visibility number $mu_{rm t}(G)$ of $G$. Graphs with $mu_{rm t}(G) = 0$ are characterized as the graphs in which no vertex is the central vertex of a convex $P_3$. The total mutual-visibility number of Cartesian products is bounded and several exact results proved. For instance, $mu_{rm t}(K_n,square, K_m) = max{n,m}$ and $mu_{rm t}(T,square, H) = mu_{rm t}(T)mu_{rm t}(H)$, where $T$ is a tree and $H$ an arbitrary graph. It is also demonstrated that $mu_{rm t}(G,square, H)$ can be arbitrary larger than $mu_{rm t}(G)mu_{rm t}(H)$.
{"title":"Graphs with total mutual-visibility number zero and total mutual-visibility in Cartesian products","authors":"S. Klavžar, Jing Tian","doi":"10.7151/dmgt.2496","DOIUrl":"https://doi.org/10.7151/dmgt.2496","url":null,"abstract":"If $G$ is a graph and $Xsubseteq V(G)$, then $X$ is a total mutual-visibility set if every pair of vertices $x$ and $y$ of $G$ admits a shortest $x,y$-path $P$ with $V(P) cap X subseteq {x,y}$. The cardinality of a largest total mutual-visibility set of $G$ is the total mutual-visibility number $mu_{rm t}(G)$ of $G$. Graphs with $mu_{rm t}(G) = 0$ are characterized as the graphs in which no vertex is the central vertex of a convex $P_3$. The total mutual-visibility number of Cartesian products is bounded and several exact results proved. For instance, $mu_{rm t}(K_n,square, K_m) = max{n,m}$ and $mu_{rm t}(T,square, H) = mu_{rm t}(T)mu_{rm t}(H)$, where $T$ is a tree and $H$ an arbitrary graph. It is also demonstrated that $mu_{rm t}(G,square, H)$ can be arbitrary larger than $mu_{rm t}(G)mu_{rm t}(H)$.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44743089","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}
Vertex-fault-tolerance was introduced by Hayes~cite{Hayes1976} in 1976, and since then it has been systematically studied in different aspects. In this paper we study $k$-vertex-fault-tolerant graphs for $p$ disjoint complete graphs of order $c$, i.e., graphs in which removing any $k$ vertices leaves a graph that has $p$ disjoint complete graphs of order $c$ as a subgraph. The main contribution is to describe such graphs that have the smallest possible number of edges for $k=1$, $p geq 1$, and $c geq 3$. Moreover, we analyze some properties of such graphs for any value of $k$.
{"title":"$k$-fault-tolerant graphs for $p$ disjoint complete graphs of order $c$","authors":"S. Cichacz, Agnieszka Gőrlich, Karol Suchan","doi":"10.7151/dmgt.2504","DOIUrl":"https://doi.org/10.7151/dmgt.2504","url":null,"abstract":"Vertex-fault-tolerance was introduced by Hayes~cite{Hayes1976} in 1976, and since then it has been systematically studied in different aspects. In this paper we study $k$-vertex-fault-tolerant graphs for $p$ disjoint complete graphs of order $c$, i.e., graphs in which removing any $k$ vertices leaves a graph that has $p$ disjoint complete graphs of order $c$ as a subgraph. The main contribution is to describe such graphs that have the smallest possible number of edges for $k=1$, $p geq 1$, and $c geq 3$. Moreover, we analyze some properties of such graphs for any value of $k$.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41661765","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}
Jaume Martí-Farré, M. Mora, M. L. Puertas, José Luis Ruiz
Abstract A dominating set of a graph is a vertex subset such that every vertex not in the subset is adjacent to at least one in the subset. In this paper we study whenever there exists a new dominating set contained (respectively, containing) the subset obtained by removing a common vertex from the union of two minimal dominating sets. A complete description of the graphs satisfying such elimination properties is provided.
{"title":"Elimination Properties for Minimal Dominating Sets of Graphs","authors":"Jaume Martí-Farré, M. Mora, M. L. Puertas, José Luis Ruiz","doi":"10.7151/dmgt.2354","DOIUrl":"https://doi.org/10.7151/dmgt.2354","url":null,"abstract":"Abstract A dominating set of a graph is a vertex subset such that every vertex not in the subset is adjacent to at least one in the subset. In this paper we study whenever there exists a new dominating set contained (respectively, containing) the subset obtained by removing a common vertex from the union of two minimal dominating sets. A complete description of the graphs satisfying such elimination properties is provided.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48876858","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}
Abstract A path factor is a spanning subgraph F of G such that every component of F is a path with at least two vertices. We write P≥k = {Pi : i ≥ k}. Then a P≥k-factor of G means a path factor in which every component admits at least k vertices, where k ≥ 2 is an integer. A graph G is called a P≥k-factor avoidable graph if for any e ∈ E(G), G admits a P≥k-factor excluding e. A graph G is called a (P≥k, n)-factor critical avoidable graph if for any Q ⊆ V (G) with |Q| = n, G − Q is a P ≥k-factor avoidable graph. Let G be an (n + 2)-connected graph. In this paper, we demonstrate that (i) G is a (P≥2, n)-factor critical avoidable graph if tough(G)>n+24 toughleft( G right) > {{n + 2} over 4} ; (ii) G is a (P≥3, n)-factor critical avoidable graph if tough(G)>n+12 toughleft( G right) > {{n + 1} over 2} ; (iii) G is a (P≥2, n)-factor critical avoidable graph if I(G)>n+23 Ileft( G right) > {{n + 2} over 3} ; (iv) G is a (P≥3, n)-factor critical avoidable graph if I(G)>n+32 Ileft( G right) > {{n + 3} over 2} . Furthermore, we claim that these conditions are sharp.
{"title":"Some Results on Path-Factor Critical Avoidable Graphs","authors":"Sizhong Zhou","doi":"10.7151/dmgt.2364","DOIUrl":"https://doi.org/10.7151/dmgt.2364","url":null,"abstract":"Abstract A path factor is a spanning subgraph F of G such that every component of F is a path with at least two vertices. We write P≥k = {Pi : i ≥ k}. Then a P≥k-factor of G means a path factor in which every component admits at least k vertices, where k ≥ 2 is an integer. A graph G is called a P≥k-factor avoidable graph if for any e ∈ E(G), G admits a P≥k-factor excluding e. A graph G is called a (P≥k, n)-factor critical avoidable graph if for any Q ⊆ V (G) with |Q| = n, G − Q is a P ≥k-factor avoidable graph. Let G be an (n + 2)-connected graph. In this paper, we demonstrate that (i) G is a (P≥2, n)-factor critical avoidable graph if tough(G)>n+24 toughleft( G right) > {{n + 2} over 4} ; (ii) G is a (P≥3, n)-factor critical avoidable graph if tough(G)>n+12 toughleft( G right) > {{n + 1} over 2} ; (iii) G is a (P≥2, n)-factor critical avoidable graph if I(G)>n+23 Ileft( G right) > {{n + 2} over 3} ; (iv) G is a (P≥3, n)-factor critical avoidable graph if I(G)>n+32 Ileft( G right) > {{n + 3} over 2} . Furthermore, we claim that these conditions are sharp.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46678559","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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