Indonesian Journal of Combinatorics最新文献

{"title":"Another H-super magic decompositions of the lexicographic product of graphs","authors":"H. Hendy, K. Sugeng, A. Salman, Nisa Ayunda","doi":"10.19184/IJC.2018.2.2.2","DOIUrl":"https://doi.org/10.19184/IJC.2018.2.2.2","url":null,"abstract":"<p>Let <span class=\"math\"><em>H</em></span> and <span class=\"math\"><em>G</em></span> be two simple graphs. The concept of an <span class=\"math\"><em>H</em></span>-magic decomposition of <span class=\"math\"><em>G</em></span> arises from the combination between graph decomposition and graph labeling. A decomposition of a graph <span class=\"math\"><em>G</em></span> into isomorphic copies of a graph <span class=\"math\"><em>H</em></span> is <span class=\"math\"><em>H</em></span>-magic if there is a bijection <span class=\"math\"><em>f</em> : <em>V</em>(<em>G</em>) ∪ <em>E</em>(<em>G</em>) → {1, 2, ..., ∣<em>V</em>(<em>G</em>) ∪ <em>E</em>(<em>G</em>)∣}</span> such that the sum of labels of edges and vertices of each copy of <span class=\"math\"><em>H</em></span> in the decomposition is constant. A lexicographic product of two graphs <span class=\"math\"><em>G</em>₁</span> and <span class=\"math\"><em>G</em>₂, </span> denoted by <span class=\"math\"><em>G</em>₁[<em>G</em>₂], </span> is a graph which arises from <span class=\"math\"><em>G</em>₁</span> by replacing each vertex of <span class=\"math\"><em>G</em>₁</span> by a copy of the <span class=\"math\"><em>G</em>₂</span> and each edge of <span class=\"math\"><em>G</em>₁</span> by all edges of the complete bipartite graph <span class=\"math\"><em>K</em>_{<em>n</em>, <em>n</em>}</span> where <span class=\"math\"><em>n</em></span> is the order of <span class=\"math\"><em>G</em>₂.</span> In this paper we provide a sufficient condition for <span class=\"math\">$overline{C_{n}}[overline{K_{m}}]$</span> in order to have a <span class=\"math\">$P_{t}[overline{K_{m}}]$</span>-magic decompositions, where <span class=\"math\"><em>n</em> > 3, <em>m</em> > 1, </span> and <span class=\"math\"><em>t</em> = 3, 4, <em>n</em> − 2</span>.</p>","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80506203","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 the local metric dimension of t-fold wheel, Pn o Km, and generalized fan","authors":"Rokhana Ayu Solekhah, T. A. Kusmayadi","doi":"10.19184/IJC.2018.2.2.4","DOIUrl":"https://doi.org/10.19184/IJC.2018.2.2.4","url":null,"abstract":"<p>Let <span class=\"math\"><em>G</em></span> be a connected graph and let <span class=\"math\"><em>u</em>, <em>v</em></span> <span class=\"math\"> ∈ </span> <span class=\"math\"><em>V</em>(<em>G</em>)</span>. For an ordered set <span class=\"math\"><em>W</em> = {<em>w</em>₁, <em>w</em>₂, ..., <em>w</em>_<em>n</em>}</span> of <span class=\"math\"><em>n</em></span> distinct vertices in <span class=\"math\"><em>G</em></span>, the representation of a vertex <span class=\"math\"><em>v</em></span> of <span class=\"math\"><em>G</em></span> with respect to <span class=\"math\"><em>W</em></span> is the <span class=\"math\"><em>n</em></span>-vector <span class=\"math\"><em>r</em>(<em>v</em>∣<em>W</em>) = (<em>d</em>(<em>v</em>, <em>w</em>₁), <em>d</em>(<em>v</em>, <em>w</em>₂), ..., </span> <span class=\"math\"><em>d</em>(<em>v</em>, <em>w</em>_<em>n</em>))</span>, where <span class=\"math\"><em>d</em>(<em>v</em>, <em>w</em>_<em>i</em>)</span> is the distance between <span class=\"math\"><em>v</em></span> and <span class=\"math\"><em>w</em>_<em>i</em></span> for <span class=\"math\">1 ≤ <em>i</em> ≤ <em>n</em></span>. The set <span class=\"math\"><em>W</em></span> is a local metric set of <span class=\"math\"><em>G</em></span> if <span class=\"math\"><em>r</em>(<em>u</em> ∣ <em>W</em>) ≠ <em>r</em>(<em>v</em> ∣ <em>W</em>)</span> for every pair <span class=\"math\"><em>u</em>, <em>v</em></span> of adjacent vertices of <span class=\"math\"><em>G</em></span>. The local metric set of <span class=\"math\"><em>G</em></span> with minimum cardinality is called a local metric basis for <span class=\"math\"><em>G</em></span> and its cardinality is called a local metric dimension, denoted by <span class=\"math\"><em>l</em><em>m</em><em>d</em>(<em>G</em>)</span>. In this paper we determine the local metric dimension of a <span class=\"math\"><em>t</em></span>-fold wheel graph, <span class=\"math\"><em>P</em>_<em>n</em></span> <span class=\"math\"> ⊙ </span> <span class=\"math\"><em>K</em>_<em>m</em></span> graph, and generalized fan graph.</p>","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84683913","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":"Some methods for constructing some classes of graceful uniform trees","authors":"I. N. Suparta, I. D. M. A. Ariawan","doi":"10.19184/ijc.2018.2.2.7","DOIUrl":"https://doi.org/10.19184/ijc.2018.2.2.7","url":null,"abstract":"<p>A tree <span class=\"math\"><em>T</em>(<em>V</em>, <em>E</em>)</span> is <span><em>graceful</em></span> if there exists an injective function <span class=\"math\"><em>f</em></span> from the vertex set <span class=\"math\"><em>V</em>(<em>T</em>)</span> into the set <span class=\"math\">{0, 1, 2, ..., ∣<em>V</em>∣ − 1}</span> which induces a bijective function <span class=\"math\"><em>f</em>ʹ</span> from the edge set <span class=\"math\"><em>E</em>(<em>T</em>)</span> onto the set <span class=\"math\">{1, 2, ..., ∣<em>E</em>∣}</span>, with <span class=\"math\"><em>f</em>ʹ(<em>u</em><em>v</em>) = ∣<em>f</em>(<em>u</em>) − <em>f</em>(<em>v</em>)∣</span> for every edge <span class=\"math\">{<em>u</em>, <em>v</em>} ∈ <em>E</em></span>. Motivated by the conjecture of Alexander Rosa (see) saying that all trees are graceful, a lot of works have addressed gracefulness of some trees. In this paper we show that some uniform trees are graceful. This results will extend the list of graceful trees.</p>","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84055341","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 star coloring of Mycielskians","authors":"K. Kaliraj, V. Kowsalya, Vernold Vivin","doi":"10.19184/IJC.2018.2.2.3","DOIUrl":"https://doi.org/10.19184/IJC.2018.2.2.3","url":null,"abstract":"<p>In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph <span class=\"math\"><em>G</em></span> into a new graph <span class=\"math\"><em>μ</em>(<em>G</em>)</span>, we now call the Mycielskian of <span class=\"math\"><em>G</em></span>, which has the same clique number as <span class=\"math\"><em>G</em></span> and whose chromatic number equals <span class=\"math\"><em>χ</em>(<em>G</em>) + 1</span>. In this paper, we find the star chromatic number for the Mycielskian graph of complete graphs, paths, cycles and complete bipartite graphs.</p>","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"91 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83028421","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":"Z2nm-supermagic labeling of Cn#Cm","authors":"D. Froncek, James McKeown, John McKeown, Michael McKeown","doi":"10.19184/IJC.2018.2.2.1","DOIUrl":"https://doi.org/10.19184/IJC.2018.2.2.1","url":null,"abstract":"<p>A <span><span class=\"math\">Γ</span>-supermagic labeling</span> of a graph <span class=\"math\"><em>G</em> = (<em>V</em>, <em>E</em>)</span> with <span class=\"math\">∣<em>E</em>∣ = <em>k</em></span> is a bijection from <span class=\"math\"><em>E</em></span> to an Abelian group <span class=\"math\">Γ</span> of order <span class=\"math\"><em>k</em></span> such that the sum of labels of all incident edges of every vertex <span class=\"math\"><em>x</em> ∈ <em>V</em></span> is equal to the same element <span class=\"math\"><em>μ</em> ∈ Γ</span>. We present a <span class=\"math\"><em>Z</em>_{2<em>n</em><em>m</em>}</span>-supermagic labeling of Cartesian product of two cycles, <span class=\"math\"><em>C</em>_<em>n</em>□<em>C</em>_<em>m</em></span> for <span class=\"math\"><em>n</em></span> odd. This along with an earlier result by Ivančo proves that a <span class=\"math\"><em>Z</em>_{2<em>n</em><em>m</em>}</span>-supermagic labeling of <span class=\"math\"><em>C</em>_<em>n</em>□<em>C</em>_<em>m</em></span> exists for every <span class=\"math\"><em>n</em>, <em>m</em> ≥ 3</span>.</p>","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79845499","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":"Edge magic total labeling of lexicographic product C4(2r+1) o ~K2 cycle with chords, unions of paths, and unions of cycles and paths","authors":"Inne Singgih","doi":"10.19184/IJC.2018.2.2.6","DOIUrl":"https://doi.org/10.19184/IJC.2018.2.2.6","url":null,"abstract":"<p>An <em>edge magic total (EMT) labeling</em> of a graph <span class=\"math\"><em>G</em> = (<em>V</em>, <em>E</em>)</span> is a bijection from the set of vertices and edges to a set of numbers defined by <span class=\"math\"><em>λ</em> : <em>V</em> ∪ <em>E</em> → {1, 2, ..., ∣<em>V</em>∣ + ∣<em>E</em>∣}</span> with the property that for every <span class=\"math\"><em>x</em><em>y</em> ∈ <em>E</em></span>, the weight of <span class=\"math\"><em>x</em><em>y</em></span> equals to a constant <span class=\"math\"><em>k</em></span>, that is, <span class=\"math\"><em>λ</em>(<em>x</em>) + <em>λ</em>(<em>y</em>) + <em>λ</em>(<em>x</em><em>y</em>) = <em>k</em></span> for some integer <span class=\"math\"><em>k</em></span>. In this paper given the construction of an EMT labeling for certain lexicographic product <span class=\"math\">$C_{4(2r+1)}circ overline{K_2}$</span>, cycle with chords <span class=\"math\"><em></em>^{[<em>c</em>]<em>t</em>}<em>C</em>_<em>n</em></span>, unions of paths <span class=\"math\"><em>m</em><em>P</em>_<em>n</em></span>, and unions of cycles and paths <span class=\"math\"> <em>m</em>(<em>C</em>_{<em>n</em>₁(2<em>r</em> + 1)} ∪ (2<em>r</em> + 1)<em>P</em>_{<em>n</em>₂})</span>.</p>","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78893870","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":"Application of generalised hierarchical product of graphs for computing F-index of four operations on graphs","authors":"Nilanjan De","doi":"10.19184/IJC.2018.2.2.5","DOIUrl":"https://doi.org/10.19184/IJC.2018.2.2.5","url":null,"abstract":"The F-index of a graph is defined as the sum of cubes of the vertex degrees of the graph which was introduced in 1972, in the same paper where the first and second Zagreb indices were introduced. In this paper we study the F-index of four operations on graphs which were introduced by Eliasi and Taeri, and hence using the derived results we find F-index of some particular and chemically interesting graphs.","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"147 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72679382","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":"Graceful labeling on torch graph","authors":"J. M. Manulang, K. Sugeng","doi":"10.19184/IJC.2018.2.1.2","DOIUrl":"https://doi.org/10.19184/IJC.2018.2.1.2","url":null,"abstract":"Let G be a graph with vertex set V=V(G) and edge set E=E(G). An injective function f:V<span style=\"font-family: symbol;\"> --> </span>{0,1,2,...,|E|} is called graceful labeling if f induces a function f^*(uv)=|f(u)<span style=\"font-family: symbol;\">-</span>f(v)| which is a bijection from E(G) to the set {1,2,3,...,|E|}. A graph which admits a graceful labeling is called a graceful graph. In this paper, we show that torch graph O_n is a graceful graph.","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80119383","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":"Further Results on Locating Chromatic Number for Amalgamation of Stars Linking by One Path","authors":"A. Asmiati, L. Yulianti, C.Ike Widyastuti","doi":"10.19184/IJC.2018.2.1.6","DOIUrl":"https://doi.org/10.19184/IJC.2018.2.1.6","url":null,"abstract":"Let G = (V,E) be a connected graph. Let c be a proper coloring using k colors, namely 1, 2,·s, k. Let <span style=\"font-family: symbol;\">P</span>={S₁, S₂,..., S_k} be a partition of V(G) induced by c and let S_i be the color class that receives the color i. The color code, c_{<span style=\"font-family: symbol;\">P</span>}(v)=(d(v,S₁), d(v,S₂),...,d(v,S_k)), where d(v,S_i)=min {d(v,x)|x <span style=\"font-family: symbol;\">Î</span> S_i} for i <span style=\"font-family: symbol;\">Î</span> [1,k]. If all vertices in V(G) have different color codes, then c is called as the emphlocating-chromatic k-coloring of G. Minimum k such that G has the locating-chromatic k-coloring is called the locating-chromatic number, denoted by <span style=\"font-family: symbol;\">c</span>_L(G). In this paper, we discuss the locating-chromatic number for n certain amalgamation of stars linking a path, denoted by nS_k,m, for n ≥ 1, m ≥ 2, k ≥ 3, and k>m.","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82360831","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":"3-Difference cordial labeling of some path related graphs","authors":"R. Ponraj, M. M. Adaickalam, R. Kala","doi":"10.19184/ijc.2018.2.1.1","DOIUrl":"https://doi.org/10.19184/ijc.2018.2.1.1","url":null,"abstract":"Let G be a (p, q) graph. Let f : V (G) → {1, 2, . . . , k} be a map where k is an integer 2 ≤ k ≤ p. For each edge uv, assign the label |f(u) − f(v)|. f is called k-difference cordial labeling of G if |vf (i) − vf (j)| ≤ 1 and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labelled with x, ef (1) and ef (0) respectively denote the number of edges labelled with 1 and not labelled with 1. A graph with a k-difference cordial labeling is called a k-difference cordial graph. In this paper we investigate 3-difference cordial labeling behavior of triangular snake, alternate triangular snake, alternate quadrilateral snake, irregular triangular snake, irregular quadrilateral snake, double triangular snake, double quadrilateral snake, double alternate triangular snake, and double alternate quadrilateral snake.","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86159995","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}