Given a graph $G$ and assuming that some vertices of $G$ are infected, the $r$-neighbor bootstrap percolation rule makes an uninfected vertex $v$ infected if $v$ has at least $r$ infected neighbors. The $r$-percolation number, $m(G, r)$, of $G$ is the minimum cardinality of a set of initially infected vertices in $G$ such that after continuously performing the $r$-neighbor bootstrap percolation rule each vertex of $G$ eventually becomes infected. In this paper, we consider the $3$-bootstrap percolation number of grids with fixed widths. If $G$ is the cartesian product $P_3 square P_m$ of two paths of orders~$3$ and $m$, we prove that $m(G,3)=frac{3}{2}(m+1)-1$, when $m$ is odd, and $m(G,3)=frac{3}{2}m +1$, when $m$ is even. Moreover if $G$ is the cartesian product $P_5 square P_m$, we prove that $m(G,3)=2m+2$, when $m$ is odd, and $m(G,3)=2m+3$, when $m$ is even. If $G$ is the cartesian product $P_4 square P_m$, we prove that $m(G,3)$ takes on one of two possible values, namely $m(G,3) = lfloor frac{5(m+1)}{3} rfloor + 1$ or $m(G,3) = lfloor frac{5(m+1)}{3} rfloor + 2$.
{"title":"3-Neighbor bootstrap percolation on grids","authors":"Jaka Hedžet, Michael A. Henning","doi":"10.7151/dmgt.2531","DOIUrl":"https://doi.org/10.7151/dmgt.2531","url":null,"abstract":"Given a graph $G$ and assuming that some vertices of $G$ are infected, the $r$-neighbor bootstrap percolation rule makes an uninfected vertex $v$ infected if $v$ has at least $r$ infected neighbors. The $r$-percolation number, $m(G, r)$, of $G$ is the minimum cardinality of a set of initially infected vertices in $G$ such that after continuously performing the $r$-neighbor bootstrap percolation rule each vertex of $G$ eventually becomes infected. In this paper, we consider the $3$-bootstrap percolation number of grids with fixed widths. If $G$ is the cartesian product $P_3 square P_m$ of two paths of orders~$3$ and $m$, we prove that $m(G,3)=frac{3}{2}(m+1)-1$, when $m$ is odd, and $m(G,3)=frac{3}{2}m +1$, when $m$ is even. Moreover if $G$ is the cartesian product $P_5 square P_m$, we prove that $m(G,3)=2m+2$, when $m$ is odd, and $m(G,3)=2m+3$, when $m$ is even. If $G$ is the cartesian product $P_4 square P_m$, we prove that $m(G,3)$ takes on one of two possible values, namely $m(G,3) = lfloor frac{5(m+1)}{3} rfloor + 1$ or $m(G,3) = lfloor frac{5(m+1)}{3} rfloor + 2$.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139354454","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}
S. Alikhani, D. Bakhshesh, H. Golmohammadi, E. Konstantinova
The connected coalition in a graph $G=(V,E)$ consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a connected dominating set but whose union $V_{1}cup V_{2}$, is a connected dominating set. A connected coalition partition in a graph $G$ of order $n=|V|$ is a vertex partition $psi$ = ${V_1, V_2,..., V_k }$ such that every set $V_i in psi$ either is a connected dominating set consisting of a single vertex of degree $n-1$, or is not a connected dominating set but forms a connected coalition with another set $V_jin psi$ which is not a connected dominating set. The connected coalition number, denoted by $CC(G)$, is the maximum cardinality of a connected coalition partition of $G$. In this paper, we initiate the study of connected coalition in graphs and present some basic results. Precisely, we characterize all graphs that have a connected coalition partition. Moreover, we show that for any graph $G$ of order $n$ with $delta(G)=1$ and with no full vertex, it holds that $CC(G)
{"title":"Connected coalitions in graphs","authors":"S. Alikhani, D. Bakhshesh, H. Golmohammadi, E. Konstantinova","doi":"10.7151/dmgt.2509","DOIUrl":"https://doi.org/10.7151/dmgt.2509","url":null,"abstract":"The connected coalition in a graph $G=(V,E)$ consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a connected dominating set but whose union $V_{1}cup V_{2}$, is a connected dominating set. A connected coalition partition in a graph $G$ of order $n=|V|$ is a vertex partition $psi$ = ${V_1, V_2,..., V_k }$ such that every set $V_i in psi$ either is a connected dominating set consisting of a single vertex of degree $n-1$, or is not a connected dominating set but forms a connected coalition with another set $V_jin psi$ which is not a connected dominating set. The connected coalition number, denoted by $CC(G)$, is the maximum cardinality of a connected coalition partition of $G$. In this paper, we initiate the study of connected coalition in graphs and present some basic results. Precisely, we characterize all graphs that have a connected coalition partition. Moreover, we show that for any graph $G$ of order $n$ with $delta(G)=1$ and with no full vertex, it holds that $CC(G)<n$. Furthermore, we show that for any tree $T$, $CC(T)=2$. Finally, we present two polynomial-time algorithms that for a given connected graph $G$ of order $n$ determine whether $CC(G)=n$ or $CC(G)=n-1$.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43501893","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 finite, simple, isolate-free graph. Two disjoint sets $A,Bsubset V$ form a total coalition in $G$, if none of them is a total dominating set, but their union $Acup B$ is a total dominating set. A vertex partition $Psi={C_1,C_2,dots,C_k}$ is a total coalition partition, if none of the partition classes is a total dominating set, meanwhile for every $iin{1,2,dots,k}$ there exists a distinct $jin{1,2,dots,k}$ such that $C_i$ and $C_j$ form a total coalition. The maximum cardinality of a total coalition partition of $G$ is the total coalition number of $G$ and denoted by $TC(G)$. We give a general sharp upper bound on the total coalition number as a function of the maximum degree. We further investigate this optimal case and study the total coalition graph. We show that every graph can be realised as a total coalition graph.
{"title":"General sharp upper bounds on the total coalition number","authors":"J'anos Bar'at, Zolt'an L. Bl'azsik","doi":"10.7151/dmgt.2511","DOIUrl":"https://doi.org/10.7151/dmgt.2511","url":null,"abstract":"Let $G(V,E)$ be a finite, simple, isolate-free graph. Two disjoint sets $A,Bsubset V$ form a total coalition in $G$, if none of them is a total dominating set, but their union $Acup B$ is a total dominating set. A vertex partition $Psi={C_1,C_2,dots,C_k}$ is a total coalition partition, if none of the partition classes is a total dominating set, meanwhile for every $iin{1,2,dots,k}$ there exists a distinct $jin{1,2,dots,k}$ such that $C_i$ and $C_j$ form a total coalition. The maximum cardinality of a total coalition partition of $G$ is the total coalition number of $G$ and denoted by $TC(G)$. We give a general sharp upper bound on the total coalition number as a function of the maximum degree. We further investigate this optimal case and study the total coalition graph. We show that every graph can be realised as a total coalition graph.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48067242","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}
This paper completes our studies on the Ramsey number r(Tn, G) for trees Tn of order n and connected graphs G of order six. If χ(G) ≥ 4, then the values of r(Tn, G) are already known for any tree Tn. Moreover, r(Sn, G), where Sn denotes the star of order n, has been investigated in case of χ(G) ≤ 3. If χ(G) = 3 and G 6= K2,2,2, then r(Sn, G) has been determined except for some G and some small n. Partial results have been obtained for r(Sn,K2,2,2) and for r(Sn, G) with χ(G) = 2. In the present paper we investigate r(Tn, G) for non-star trees Tn and χ(G) ≤ 3. Especially, r(Tn, G) is completely evaluated for any non-star tree Tn if χ(G) = 3 where G 6= K2,2,2, and r(Tn,K2,2,2) is determined for a class of trees Tn with small maximum degree. In case of χ(G) = 2, r(Tn, G) is investigated for Tn = Pn, the path of order n, and for Tn = B2,n−2, the special broom of order n obtained by identifying the centre of a star S3 with an end-vertex of a path Pn−2. Furthermore, the values of r(B2,n−2, Sm) are determined for all n and m with n ≥ m− 1. As a consequence of this paper, r(F,G) is known for all trees F of order at most five and all connected graphs G of order at most six.
{"title":"On the Ramsey numbers of non-star trees versus connected graphs of order six","authors":"Roland Lortz, I. Mengersen","doi":"10.7151/dmgt.2370","DOIUrl":"https://doi.org/10.7151/dmgt.2370","url":null,"abstract":"This paper completes our studies on the Ramsey number r(Tn, G) for trees Tn of order n and connected graphs G of order six. If χ(G) ≥ 4, then the values of r(Tn, G) are already known for any tree Tn. Moreover, r(Sn, G), where Sn denotes the star of order n, has been investigated in case of χ(G) ≤ 3. If χ(G) = 3 and G 6= K2,2,2, then r(Sn, G) has been determined except for some G and some small n. Partial results have been obtained for r(Sn,K2,2,2) and for r(Sn, G) with χ(G) = 2. In the present paper we investigate r(Tn, G) for non-star trees Tn and χ(G) ≤ 3. Especially, r(Tn, G) is completely evaluated for any non-star tree Tn if χ(G) = 3 where G 6= K2,2,2, and r(Tn,K2,2,2) is determined for a class of trees Tn with small maximum degree. In case of χ(G) = 2, r(Tn, G) is investigated for Tn = Pn, the path of order n, and for Tn = B2,n−2, the special broom of order n obtained by identifying the centre of a star S3 with an end-vertex of a path Pn−2. Furthermore, the values of r(B2,n−2, Sm) are determined for all n and m with n ≥ m− 1. As a consequence of this paper, r(F,G) is known for all trees F of order at most five and all connected graphs G of order at most six.","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":"90363430","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":"On non-hamiltonian polyhedra without cubic vertices and their vertex-deleted subgraphs","authors":"Jan Goedgebeur, Thomas Gringore, Carol Zamfirescu","doi":"10.7151/dmgt.2514","DOIUrl":"https://doi.org/10.7151/dmgt.2514","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":"135442100","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}
Deise L. de Oliveira, Danilo Artigas, S. Dantas, A. G. Luiz
{"title":"On the edge-sum distinguishing game","authors":"Deise L. de Oliveira, Danilo Artigas, S. Dantas, A. G. Luiz","doi":"10.7151/dmgt.2502","DOIUrl":"https://doi.org/10.7151/dmgt.2502","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":"71130043","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 Caro, Shapira and Yuster [1] it is proven that for any graph G with at least 5 vertices, one can delete at most 6 vertices such that the subgraph obtained has at least three vertices with the same degree. Furthermore they show that for certain graphs one needs to remove at least 3 vertices in order that the resulting graph has at least 3 vertices of the same degree. In this note we prove that for any graph G with at least 5 vertices, one can delete at most 5 vertices such that the subgraph obtained has at least three vertices with the same degree. We also show that for any triangle-free graph G with at least 6 vertices, one can delete at most one vertex such that the subgraph obtained has at least three vertices with the same degree and this result is tight for triangle-free graphs.
{"title":"A note on forcing 3-repetitions in degree sequences","authors":"Shimon Kogan","doi":"10.7151/dmgt.2376","DOIUrl":"https://doi.org/10.7151/dmgt.2376","url":null,"abstract":"In Caro, Shapira and Yuster [1] it is proven that for any graph G with at least 5 vertices, one can delete at most 6 vertices such that the subgraph obtained has at least three vertices with the same degree. Furthermore they show that for certain graphs one needs to remove at least 3 vertices in order that the resulting graph has at least 3 vertices of the same degree. In this note we prove that for any graph G with at least 5 vertices, one can delete at most 5 vertices such that the subgraph obtained has at least three vertices with the same degree. We also show that for any triangle-free graph G with at least 6 vertices, one can delete at most one vertex such that the subgraph obtained has at least three vertices with the same degree and this result is tight for triangle-free 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":"91337043","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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