{"title":"Chromatics Number of Operation Graphs","authors":"Kiki Kurdianto, Ika Hesti Agustin, D. Dafik","doi":"10.25037/cgantjma.v1i1.9","DOIUrl":"https://doi.org/10.25037/cgantjma.v1i1.9","url":null,"abstract":"Let G = (V (G); E(G)) be connected nontrivial graph. Edge coloring is de-¯ned as c : E(G) ! f1; 2; :::; kg; k 2 N, with the conditions no edges adja-cent having the same color. Coloring k-color edges r-dynamic is edges color-ing as much as k color such that every edges in E(G) with adjacent at least minfr; d(u) + d(v) ¡ 2g have di®erent color. An Edge r dynamic is a proper c of E(G) such that jc(N(uv))j = minfr; d(u) + d(v) ¡ 2g, for each edge N(uv) is the neighborhood of uv and c(N(uv)) is color used to with adjacent edges of uv. the edge r-dynamic chromatic number, written as ¸(G), is the minimum k such that G has an edge r-dynamic k-coloring. chromatic number 1-dynamic writ-ten as ¸(G), chromatic number 2-dynamic written as ¸d(G) And for chromatic number r-dynamic written as ¸(G). A graph is used in this research namely gshack(H3; e; n), amal(Bt3; v; n) and amal(S4; v; n). Keywords: r-dynamic coloring, r-dynamic chromatic number, graph operations.","PeriodicalId":305608,"journal":{"name":"CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS","volume":"95 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131002379","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}
{"title":"Analisis Locating Dominating Set pada Graf Khusus dan Hasil Operasi Comb Sisi","authors":"Imro’atun Rofikah","doi":"10.25037/cgantjma.v1i2.40","DOIUrl":"https://doi.org/10.25037/cgantjma.v1i2.40","url":null,"abstract":"Assume that G = (V;E) is an undirected and connected graph with vertex set V and edge set E. D is called a dominating set of the vertex in G such that for each vertex v 2 V one of: v 2 D or a neighbor u of v in D with u 2 D. While locating dominating set of G is a dominating set D of G when satisfy this condition: for every two vertices u; v 2 (V D);N(u) DN(v) D. The minimum cardinality of a locating dominating set of G is the location domination number L(G). In this paper, locating dominating set and location domination number of special graph and edge comb product operation result will be determined. Location domination number theorem on triangular book graph Btn and edge comb product operation result that is Cm D Btn and Sm D Btn are the results from this experiment.","PeriodicalId":305608,"journal":{"name":"CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123383822","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}
{"title":"On r-Dynamic Coloring for Graph Operation of Cycle, Star, Complete, and Path","authors":"D. Puspasari., D. Dafik, S. Slamin","doi":"10.25037/cgantjma.v1i1.2","DOIUrl":"https://doi.org/10.25037/cgantjma.v1i1.2","url":null,"abstract":"For integer k, r > 0, (k, r) -coloring of graph G is a proper coloring on the vertices of G by k-colors such that every vertex v of degree d(v) is adjacent to vertices with at least min{d(v), r} different color. By a proper k -coloring of graph G, we mean a map c : V (G) → S, where |S| = k, such that any two adjacent vertices are different color. An r -dynamic k -coloring is a proper k -coloring c of G such that |c(N (v))| ≥ min{r, d(v)} for each vertex v in V (G), where N (v) is the neighborhood of v and c(S) = {c(v) : v ∈ S} for a vertex subset S . The r-dynamic chromatic number, written as χr (G), is the minimum k such that G has an r-dynamic k-coloring. Note the 1-dynamic chromatic number of graph is equal to its chromatic number, denoted by χ(G), and the 2-dynamic chromatic number of graph denoted by χd (G). By simple observation with a greedy coloring algorithm, it is easy to see that χr (G) ≤ χr+1(G), however χr+1(G) − χr (G) does not always have the same difference. Thus finding an exact values of χr (G) is significantly useful. In this paper, we investigate the some exact value of χr (G) when G is for an operation product of cycle, star, complete, and path graphs.","PeriodicalId":305608,"journal":{"name":"CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132328543","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}
{"title":"Kajian Rainbow 2-Connected Pada Graf Eksponensial dan Beberapa Operasi Graf","authors":"H. Oktaviana","doi":"10.25037/cgantjma.v2i2.56","DOIUrl":"https://doi.org/10.25037/cgantjma.v2i2.56","url":null,"abstract":"Let $G=(V(G),E(G))$ is a graph connected non-trivial. textit{Rainbow connection} is edge coloring on the graph defined as $f:E(G)rightarrow {1,2,...,r|r in N}$, for every two distinct vertices in $G$ have at least one textit{rainbow path}. The graph $G$ says textit{rainbow connected} if every two vertices are different in $G$ associated with textit{rainbow path}. A path $u-v$ in $G$ says textit{rainbow path} if there are no two edges in the trajectory of the same color. The edge coloring sisi cause $G$ to be textit{rainbow connected} called textit{rainbow coloring}. Minimum coloring in a graph $G$ called textit{rainbow connection number} which is denoted by $rc(G)$. If the graph $G$ has at least two textit{disjoint rainbow path} connecting two distinct vertices in $G$. So graph $G$ is called textit{rainbow 2-connected} which is denoted by $rc_2(G)$. The purpose of this research is to determine textit{rainbow 2-connected} of some resulting graph operations. This research study textit{rainbow 2-connected} on the graph (${C_4}^{K_n}$ and $Wd_{(3,2)}square K_n$). ","PeriodicalId":305608,"journal":{"name":"CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS","volume":"73 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130769513","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}
{"title":"Analisa Antimagicness Super dari Shackle Graf Parasut dan Aplikasinya pada Polyalphabetic Cipher","authors":"Riza Nurfadila","doi":"10.25037/cgantjma.v2i1.50","DOIUrl":"https://doi.org/10.25037/cgantjma.v2i1.50","url":null,"abstract":"Super (emph{a,d})-$mathcal{H}$-antimagic total covering on a graph emph{G}=(emph{V,E}) is the total labeling of $lambda$ of emph {V(G)} $cup$ emph{E(G)} with positive integers {1, 2, 3,dots ,$|V(G) cup E(G)|$}, for any subgraph emph{H'} of emph{G} that is isomorphic to emph{H} where $sum$ emph{H'} = $sum_{v in V(H)} lambda (v ) + sum_{e in E(H)} lambda (e)$ is an arithmetic sequence {emph{a, a+d, a+2d,dots,a+(s-1)d}} where emph{a}, emph{d} are positive numbers where emph{a} is the first term, emph{d} is the difference, and emph{s} is the number of covers. If $lambda(v)_{v in V} = {1,2,3,dots,|V(G)|}$ then the graph emph{G} have the label of super $mathcal{H}$-antimagic covering. One of the techniques that can be applied to get the super antimagic total covering on the graph is the partition technique. Graph applications that can be developed for super antimagic total covering are emph{ciphertext} and emph{streamcipher}. emph{Ciphertext} is an encrypted message and is related to cryptography. emph{Stream cipher} is an extension of emph{Ciphertext}. This article study the super (a,d)-$mathcal{H}$-antimagic total covering on the shackle of parachute graph and its application in emph{ciphertext}. The graphs that used in this article are some parachute graphs denoted by emph{shack}($mathcal{P}_{m},e,n$).","PeriodicalId":305608,"journal":{"name":"CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124983467","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}
{"title":"Analisis Rainbow Vertex Connection pada Beberapa Graf Khusus dan Operasinya","authors":"Ida Ariska","doi":"10.25037/cgantjma.v2i1.53","DOIUrl":"https://doi.org/10.25037/cgantjma.v2i1.53","url":null,"abstract":"Suppose $G=(V(G),E(G))$ is a non-trivial connected graph with edge coloring defined as $c:E(G) rightarrow {1,2,...,k} ,k in N$, with the condition that neighboring edges can be the same color. An original path is {it rainbow path} if there are no two edges in the path of the same color. The graph $G$ is called rainbow connected if every two vertices in $G$ with rainbow path in $G$. The coloring here is called rainbow coloring, and the minimal coloring in a graph $G$ rainbow connection number is denoted by $rc(G)$. Suppose $G=(V(G),E(G))$ is a non-trivial connected graph with a vertex coloring defined as $c':V(G) rightarrow {1,2,...,k},k in N$, with the condition that neighboring interior vertex may have the same color. An original path is rainbow vertex path if there are no two vertices in the path of the same color. The graph $G$ is called rainbow vertex connected if every two vertices in $G$ with rainbow vertex path in $G$. The $G$ coloring is called rainbow vertex coloring, and the minimal coloring in a $G$ graph is called rainbow vertex connection number which is denoted by $rvc(G)$. This research produces rainbow vertex connection number on the graph resulting from the operation emph{amal}($Bt_{m}$, $v$, $n$), $Wd_{3,m}$ $Box$ $ P_n$, $P_m$ $odot$ $Wd_{3,n}$, $Wd_{3,m}$ $+$ $C_n$, and emph{shack}($Bt_{m}$, $v $, $n$). ","PeriodicalId":305608,"journal":{"name":"CGANT JOURNAL OF MATHEMATICS AND APPLICATIONS","volume":"258 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117014900","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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