Let G and H be two given graphs. The notation F→(G,H) means that any red-blue coloring on the edges of F will create either a red subgraph G or a blue subgraph H in F. A graph F is a Ramsey (G,H)-minimal graph if F satisfies two conditions: (1) F→(G,H), and (2) (F−e) ⇸ (G,H) for every e ∈ E(F). Denote ℜ(G,H) as the set of all (G,H)-minimal graphs. In this paper we prove that a tree T is not in ℜ(mK2,bPn) if it has a diameter of at least n(b+m−1)−1 for m,n,b≥2, furthermore we show that (b+m−1)Pn ∈ ℜ(mK2,bPn) for every m,n,b≥2. We also prove that for n≥3, a cycle on k vertices is in ℜ(mK2,bPn) if and only if k ∈ [n(b+m−2)+1, n(b+m−1)−1].
假设 G 和 H 是两个给定的图。符号 F→(G,H) 表示 F 边上的任何红蓝着色都会在 F 中创建一个红色子图 G 或蓝色子图 H。如果 F 满足以下两个条件，则图 F 是拉姆齐 (G,H) 最小图：(1) F→(G,H)；(2) 对于每个 e∈ E(F)，(F-e) ⇸ (G,H)。表示 ℜ(G,H) 是所有 (G,H) 最小图的集合。本文将证明，对于 m,n,b≥2，如果树 T 的直径至少为 n(b+m-1)-1，则该树 T 不在ℜ(mK2,bPn)中；此外，我们还将证明，对于每一个 m,n,b≥2，(b+m-1)Pn∈ ℜ(mK2,bPn)。我们还证明，对于 n≥3，当且仅当 k∈ [n(b+m-2)+1, n(b+m-1)-1] 时，k 个顶点上的循环在ℜ(mK2,bPn)中。
{"title":"On Ramsey (mK2,bPn)-minimal Graphs","authors":"Nadia Nadia, L. Yulianti, F. F. Hadiputra","doi":"10.19184/ijc.2023.7.1.2","DOIUrl":"https://doi.org/10.19184/ijc.2023.7.1.2","url":null,"abstract":"<p style=\"text-align: justify;\">Let <em>G</em> and <em>H</em> be two given graphs. The notation <em>F</em>→(<em>G,H</em>) means that any red-blue coloring on the edges of <em>F</em> will create either a red subgraph <em>G</em> or a blue subgraph <em>H</em> in <em>F</em>. A graph <em>F</em> is a Ramsey (<em>G,H</em>)-minimal graph if <em>F</em> satisfies two conditions: (1) <em>F→</em>(<em>G,H</em>), and (2) (<em>F</em>−<em>e</em>) ⇸ (<em>G,H</em>) for every <em>e </em>∈ <em>E</em>(<em>F</em>). Denote ℜ(<em>G,H</em>) as the set of all (<em>G,H</em>)-minimal graphs. In this paper we prove that a tree <em>T</em> is not in ℜ(<em>mK</em><sub>2</sub>,<em>bP</em><sub>n</sub>) if it has a diameter of at least <em>n</em>(<em>b+m</em>−1)−1 for <em>m,n,b</em>≥2, furthermore we show that (<em>b</em>+<em>m</em>−1)<em>P</em><sub>n</sub> ∈ ℜ(<em>mK</em><sub>2</sub>,<em>bP</em><sub>n</sub>) for every <em>m,n,b</em>≥2. We also prove that for <em>n</em>≥3, a cycle on <em>k</em> vertices is in ℜ(<em>mK</em><sub>2</sub>,<em>bP</em><sub>n</sub>) if and only if <em>k</em> ∈ [<em>n</em>(<em>b</em>+<em>m</em>−2)+1, <em>n</em>(<em>b</em>+<em>m</em>−1)−1].</p>","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139362527","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}
The total vertex irregularity strength of a graph G=(V,E) is the minimum integer k so that there is a mapping from V ∪ E to the set {1,2,...,k} so that the vertex-weights (i.e., the sum of labels of a vertex together with the edges incident to it) are all distinct. In this note, we present a new sufficient condition for a tree to have total vertex irregularity strength ⌈(n1+1)/2⌉, where n1 is the number of vertices of degree one in the tree.
{"title":"A note on vertex irregular total labeling of trees","authors":"Faisal Susanto, R. Simanjuntak, E. Baskoro","doi":"10.19184/ijc.2023.7.1.1","DOIUrl":"https://doi.org/10.19184/ijc.2023.7.1.1","url":null,"abstract":"The total vertex irregularity strength of a graph <em>G=</em>(<em>V,E</em>) is the minimum integer <em>k</em> so that there is a mapping from <em>V ∪ E</em> to the set {<em>1,2,...,k</em>} so that the vertex-weights (i.e., the sum of labels of a vertex together with the edges incident to it) are all distinct. In this note, we present a new sufficient condition for a tree to have total vertex irregularity strength ⌈(<em>n</em><sub>1</sub><em>+1</em>)<em>/2</em>⌉<em><em>, where <em>n<sub>1</sub></em> is the number of vertices of degree one in the tree.</em></em>","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"97 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139362499","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}
A Γ-supermagic labeling of a graph G=(V,E) is a bijection from E to a group Γ of order |E| such that the sum of labels of all edges incident with any vertex x∈ V is equal to the same element μ ∈ Γ.
A Z2mn-supermagic labeling of the Cartesian product of two cycles, Cm ℺ Cn for every m,n ≥ 3 was found by Froncek, McKeown, McKeown, and McKeown. In this paper we present a Zk-supermagic labeling of the direct and strong product by cyclic group Zk for any m,n ≥ 3.
{"title":"Γ-supermagic labeling of products of two cycles with cyclic groups","authors":"D. Froncek","doi":"10.19184/ijc.2023.7.1.3","DOIUrl":"https://doi.org/10.19184/ijc.2023.7.1.3","url":null,"abstract":"A <em>Γ</em>-supermagic labeling of a graph <em>G</em>=(<em>V,E</em>) is a bijection from <em>E</em> to a group <em>Γ</em> of order |<em>E</em>| such that the sum of labels of all edges incident with any vertex <em>x</em>∈ <em>V</em> is equal to the same element μ ∈ <em>Γ</em>. <br /><br />A <em>Z</em><sub>2mn</sub>-supermagic labeling of the Cartesian product of two cycles, <em>C</em><sub>m</sub> ℺ <em>C</em><sub>n</sub> for every <em>m,n</em> ≥ 3 was found by Froncek, McKeown, McKeown, and McKeown. In this paper we present a <em>Z</em><sub>k</sub>-supermagic labeling of the direct and strong product by cyclic group <em>Z</em><sub>k</sub> for any <em>m,n</em> ≥ 3.","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139362749","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}
A rainbow geodesic is a shortest path between two vertices where all edges are colored differently. An edge coloring in which any pair of vertices with distance up to d, where d is a positive integer that can be connected by a rainbow geodesic is called d-local strong rainbow coloring. The d-local strong rainbow connection number, denoted by lsrcd(G), is the least number of colors used in d-local strong rainbow coloring. Suppose that G and H are graphs of order m and n, respectively. The corona product of G and H, G ⊙ H, is defined as a graph obtained by taking a copy of G and m copies of H, then connecting every vertex in the i-th copy of H to the i-th vertex of G. In this paper, we will determine the lsrcd(Cm ⊙ Cn) for d=2 and d=3.
彩虹大地线是两个顶点之间的最短路径，其中所有边的颜色都不同。在边着色中，距离不超过 d（d 为正整数）的任何一对顶点都可以通过彩虹大地线连接，这种边着色称为 d 局部强彩虹着色。d 局域强彩虹连接数用 lsrcd(G) 表示，是 d 局域强彩虹着色中使用的最少颜色数。假设 G 和 H 分别是阶数为 m 和 n 的图。本文将确定 d=2 和 d=3 时的 lsrcd(Cm ⊙ Cn)。
{"title":"Local Strong Rainbow Connection Number of Corona Product Between Cycle Graphs","authors":"Khairunnisa N. Afifah, Kiki A. Sugeng","doi":"10.19184/ijc.2023.7.1.4","DOIUrl":"https://doi.org/10.19184/ijc.2023.7.1.4","url":null,"abstract":"<p style=\"text-align: justify;\">A rainbow geodesic is a shortest path between two vertices where all edges are colored differently. An edge coloring in which any pair of vertices with distance up to <em>d</em>, where <em>d</em> is a positive integer that can be connected by a rainbow geodesic is called <em>d</em>-local strong rainbow coloring. The <em>d</em>-local strong rainbow connection number, denoted by <em>lsrc</em><sub>d</sub>(<em>G</em>), is the least number of colors used in <em>d</em>-local strong rainbow coloring. Suppose that <em>G</em> and <em>H</em> are graphs of order <em>m</em> and <em>n</em>, respectively. The corona product of <em>G</em> and <em>H</em>, <em>G </em>⊙ <em>H</em>, is defined as a graph obtained by taking a copy of <em>G</em> and <em>m</em> copies of <em>H</em>, then connecting every vertex in the <em>i</em>-th copy of <em>H</em> to the <em>i</em>-th vertex of <em>G</em>. In this paper, we will determine the <em>lsrc</em><sub>d</sub>(<em>C</em><sub>m</sub> ⊙ <em>C</em><sub>n</sub>) for <em>d</em>=2 and <em>d</em>=3.</p>","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139362557","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}
Let 𝑋 be a set and (𝑋,𝑇) be a topological space. A new type of graph on 𝑃(𝑋), namely the closure graph of 𝑇 is introduced. The closure graph denoted by Γc whose vertex set is 𝑃(𝑋) in which two distinct vertices 𝐴 and 𝐵 are adjacent if A'∩B' ⊆ (A∩B)'. In this paper, the closure graph is shown as a simple, connected graph with diameter at most two. Furthermore, the girth of the closure graph Γcof 𝑇 is three if 𝑋 contains more than one point. Also, several graph properties are studied.
{"title":"On graphs associated to topological spaces","authors":"Haval Mohammed Salih","doi":"10.19184/ijc.2023.7.1.5","DOIUrl":"https://doi.org/10.19184/ijc.2023.7.1.5","url":null,"abstract":"Let 𝑋 be a set and (𝑋,𝑇) be a topological space. A new type of graph on 𝑃(𝑋), namely the closure graph of 𝑇 is introduced. The closure graph denoted by <em>Γ<sup>c</sup></em> whose vertex set is 𝑃(𝑋) in which two distinct vertices 𝐴 and 𝐵 are adjacent if <em>A</em>'∩<em>B</em>' ⊆ (<em>A</em>∩<em>B</em>)'. In this paper, the closure graph is shown as a simple, connected graph with diameter at most two. Furthermore, the girth of the closure graph <em>Γ<sup>c</sup></em>of 𝑇 is three if 𝑋 contains more than one point. Also, several graph properties are studied.","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136369565","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}
A caterpillar is a tree obtained from a path by attaching pendant vertices. The number of caterpillars of size n is a well-known result. In this work extend this result exploring the number of caterpillars of size n together with the cardinalities of the stable sets and the diameter. Three closed formulas are presented, giving the number of caterpillars of size n with: (i) smaller stable set of cardinality k, (ii) diameter d, and (iii) diameter d and smaller stable set of cardinality k.
{"title":"On the number of caterpillars","authors":"Christian Barrientos","doi":"10.19184/ijc.2022.6.2.1","DOIUrl":"https://doi.org/10.19184/ijc.2022.6.2.1","url":null,"abstract":"<p style=\"text-align: justify;\">A caterpillar is a tree obtained from a path by attaching pendant vertices. The number of caterpillars of size <em>n</em> is a well-known result. In this work extend this result exploring the number of caterpillars of size <em>n</em> together with the cardinalities of the stable sets and the diameter. Three closed formulas are presented, giving the number of caterpillars of size <em>n</em> with: (i) smaller stable set of cardinality <em>k</em>, (ii) diameter <em>d</em>, and (iii) diameter <em>d</em> and smaller stable set of cardinality <em>k</em>.</p>","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81022466","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}
Let G be a subgroup of Sn. For x ∈ G, the index of x in G is denoted by ind x is the minimal number of 2-cycles needed to express x as a product. In this paper, we define a new kind of graph on G, namely the index graph and denoted by Γind(G). Its vertex set the set of all conjugacy classes of G and two distinct vertices x ∈ Cx and y ∈ Cy are adjacent if Gcd(ind x, ind y) 6 ≠ 1. We study some properties of this graph for the symmetric groups Sn, the alternating group An, the cyclic group Cn, the dihedral group D2n and the generalized quaternain group Q4n. In particular, we are interested in the connectedness of them.
{"title":"Index graphs of finite permutation groups","authors":"H. M. Mohammed Salih","doi":"10.19184/ijc.2022.6.2.2","DOIUrl":"https://doi.org/10.19184/ijc.2022.6.2.2","url":null,"abstract":"Let <em>G</em> be a subgroup of <em>S</em><sub>n</sub>. For <em>x ∈ G</em>, the index of <em>x</em> in <em>G</em> is denoted by <em>ind x</em> is the minimal number of 2-cycles needed to express <em>x</em> as a product. In this paper, we define a new kind of graph on <em>G</em>, namely the index graph and denoted by <em>Γ</em><sup>ind</sup><em>(G)</em>. Its vertex set the set of all conjugacy classes of <em>G</em> and two distinct vertices <em>x ∈ C</em><sub>x</sub> and <em>y ∈ C</em><sub>y</sub> are adjacent if <em>Gcd(ind x, ind y) 6 ≠ 1</em>. We study some properties of this graph for the symmetric groups <em>S</em><sub>n</sub>, the alternating group <em>A</em><sub>n</sub>, the cyclic group <em>C</em><sub>n</sub>, the dihedral group <em>D</em><sub>2n</sub> and the generalized quaternain group <em>Q</em><sub>4n</sub>. In particular, we are interested in the connectedness of them.","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"2016 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86623175","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}
H. Ramane, Ishwar Baidari, R. Jummannaver, V. V. Manjalapur, G. A. Gudodagi, A. Yalnaik, Ajith Hanagawadimath
Let $B(G)$ be the incidence matrix of a graph $G$. The row in $B(G)$ corresponding to a vertex $v$, denoted by $s(v)$ is the string which belongs to $Bbb{Z}_2^n$, a set of $n$-tuples over a field of order two. The Hamming distance between the strings $s(u)$ and $s(v)$ is the number of positions in which $s(u)$ and $s(v)$ differ. In this paper we obtain the Hamming distance between the strings generated by the incidence matrix of a graph. The sum of Hamming distances between all pairs of strings, called Hamming index of a graph is obtained.
{"title":"Hamming index of graphs with respect to its incidence matrix","authors":"H. Ramane, Ishwar Baidari, R. Jummannaver, V. V. Manjalapur, G. A. Gudodagi, A. Yalnaik, Ajith Hanagawadimath","doi":"10.19184/ijc.2022.6.2.4","DOIUrl":"https://doi.org/10.19184/ijc.2022.6.2.4","url":null,"abstract":"Let $B(G)$ be the incidence matrix of a graph $G$. The row in $B(G)$ corresponding to a vertex $v$, denoted by $s(v)$ is the string which belongs to $Bbb{Z}_2^n$, a set of $n$-tuples over a field of order two. The Hamming distance between the strings $s(u)$ and $s(v)$ is the number of positions in which $s(u)$ and $s(v)$ differ. In this paper we obtain the Hamming distance between the strings generated by the incidence matrix of a graph. The sum of Hamming distances between all pairs of strings, called Hamming index of a graph is obtained.","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87330361","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}
Rika Yanti, Gregory Benedict Tanidi, S. Saputro, E. Baskoro
Foster (1932) performed a mathematical census for all connected symmetric cubic (trivalent) graphs of order n with n ≤ 512. This census then was continued by Conder et al. (2006) and they obtained the complete list of all connected symmetric cubic graphs with order n ≤ 768. In this paper, we determine the total vertex irregularity strength of such graphs obtained by Foster. As a result, all the values of the total vertex irregularity strengths of the symmetric cubic graphs of order n from Foster census strengthen the conjecture stated by Nurdin, Baskoro, Gaos & Salman (2010), namely ⌈(n+3)/4⌉.
Foster(1932)对n≤512的所有连通的对称n阶(三价)图进行了数学普查。Conder et al.(2006)继续这个普查，他们得到了所有n阶≤768的连通对称三次图的完整列表。在本文中，我们确定了由Foster得到的这类图的顶点总不规则强度。因此，Foster人口普查的n阶对称三次图的顶点总不规则强度的所有值都加强了Nurdin, Baskoro, Gaos & Salman(2010)提出的猜想，即≤(n+3)/4≤。
{"title":"The total vertex irregularity strength of symmetric cubic graphs of the Foster's Census","authors":"Rika Yanti, Gregory Benedict Tanidi, S. Saputro, E. Baskoro","doi":"10.19184/ijc.2022.6.2.3","DOIUrl":"https://doi.org/10.19184/ijc.2022.6.2.3","url":null,"abstract":"Foster (1932) performed a mathematical census for all connected symmetric cubic (trivalent) graphs of order n with n ≤ 512. This census then was continued by Conder et al. (2006) and they obtained the complete list of all connected symmetric cubic graphs with order n ≤ 768. In this paper, we determine the total vertex irregularity strength of such graphs obtained by Foster. As a result, all the values of the total vertex irregularity strengths of the symmetric cubic graphs of order n from Foster census strengthen the conjecture stated by Nurdin, Baskoro, Gaos & Salman (2010), namely ⌈(n+3)/4⌉.","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74017498","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}
A property ℘ is defined to be a nonempty isomorphism-closed subclass of the class of all finite simple graphs. A nonempty set S of vertices of a graph G is said to be a ℘-set of G if G[S]∈ ℘. The maximum and minimum cardinalities of a ℘-set of G are denoted by M℘(G) and m℘(G), respectively. If S is a ℘-set such that its cardinality equals M℘(G) or m℘(G), we say that S is an M℘-set or an m℘-set of G, respectively. In this paper, we not only define six types of property ℘ by the using concepts of graph product and generalized graph product, but we also obtain M℘ and m℘ of product graphs in each type and characterize its M℘-set.
性质p被定义为所有有限简单图类的非空同构闭子类。如果G[S]∈p，则图G的顶点的非空集S称为G的p集。p的最大和最小基数分别用M p (G)和M p (G)表示。如果S是一个p -set使得它的基数等于M p (G)或M p (G)，我们说S是一个M p -set或G的M p -set。本文利用图积和广义图积的概念定义了六种性质的p (p)，得到了每种类型的p (p)图的p (p)和p (p)，并对其p (p)集进行了刻画。
{"title":"On generalized composed properties of generalized product graphs","authors":"Nopparat Pleanmani, S. Panma","doi":"10.19184/ijc.2022.6.2.5","DOIUrl":"https://doi.org/10.19184/ijc.2022.6.2.5","url":null,"abstract":"<p>A property ℘ is defined to be a nonempty isomorphism-closed subclass of the class of all finite simple graphs. A nonempty set <em>S</em> of vertices of a graph <em>G</em> is said to be a ℘-set of <em>G</em> if <em>G</em>[<em>S</em>]∈ ℘. The maximum and minimum cardinalities of a ℘-set of <em>G</em> are denoted by <em>M</em><sub>℘</sub>(<em>G</em>) and <em>m</em><sub>℘</sub>(<em>G</em>), respectively. If <em>S</em> is a ℘-set such that its cardinality equals <em>M</em><sub>℘</sub>(<em>G</em>) or <em>m</em><sub>℘</sub>(<em>G</em>), we say that <em>S</em> is an <em>M</em><sub>℘</sub>-set or an <em>m</em><sub>℘</sub>-set of <em>G</em>, respectively. In this paper, we not only define six types of property ℘ by the using concepts of graph product and generalized graph product, but we also obtain <em>M</em><sub>℘</sub> and <em>m</em><sub>℘</sub> of product graphs in each type and characterize its <em>M</em><sub>℘</sub>-set.</p>","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87973685","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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