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Another Hsuper magic decompositions of the lexicographic product of graphs
Hendy, H ( Universitas Pesantren Tinggi Darul Ulum ) , Sugeng, Kiki A. ( Universitas Indonesia ) , Salman, A.N.M ( Institut Teknologi Bandung ) , Ayunda, Nisa ( Universitas Pesantren Tinggi Darul Ulum )
Indonesian Journal of Combinatorics Vol 2, No 2 (2018)
Publisher : Indonesian Combinatorial Society (InaCombS)
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DOI: 10.19184/ijc.2018.2.2.2
Let H and G be two simple graphs. The concept of an Hmagic decomposition of G arises from the combination between graph decomposition and graph labeling. A decomposition of a graph G into isomorphic copies of a graph H is Hmagic if there is a bijection f : V(G) ∪ E(G) → {1, 2, ..., ∣V(G) ∪ E(G)∣} such that the sum of labels of edges and vertices of each copy of H in the decomposition is constant. A lexicographic product of two graphs G1 and G2, denoted by G1[G2], is a graph which arises from G1 by replacing each vertex of G1 by a copy of the G2 and each edge of G1 by all edges of the complete bipartite graph Kn, n where n is the order of G2. In this paper we provide a sufficient condition for $\overline{C_{n}}[\overline{K_{m}}]$ in order to have a $P_{t}[\overline{K_{m}}]$magic decompositions, where n > 3, m > 1, and t = 3, 4, n − 2.
On the Ramsey number of 4cycle versus wheel
Noviani, Enik ( Institut Teknologi Bandung ) , Baskoro, Edy Tri ( Institut Teknologi Bandung )
Indonesian Journal of Combinatorics Vol 1, No 1 (2016)
Publisher : Indonesian Combinatorial Society (InaCombS)
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DOI: 10.19184/ijc.2016.1.1.2
For any fixed graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest positive integer $n$ such that for every graph $F$ on $n$ vertices must contain $G$ or the complement of $F$ contains $H$. The girth of graph $G$ is a length of the shortest cycle. A $k$regular graph with the girth $g$ is called a $(k,g)$graph. If the number of of vertices in $(k,g)$graph is minimized then we call this graph a $(k,g)$cage. In this paper, we derive the bounds of Ramsey number $R(C_4,W_n)$ for some values of $n$. By modifying $(k, 5)$graphs, for $k = 7$ or $9$, we construct these corresponding $(C_4,W_n)$good graphs.
Zagreb indices of blockedge transformation graphs and their complements
Basavanagoud, Bommanahal ( Karnatak University ) , Patil, Shreekant ( Karnatak University )
Indonesian Journal of Combinatorics Vol 1, No 2 (2017)
Publisher : Indonesian Combinatorial Society (InaCombS)
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DOI: 10.19184/ijc.2017.1.2.3
In this paper, we obtain expressions for first and second Zagreb indices and coindices of blockedge transformation graphs G^{ab}. Analogous expressions are obtained also for the complements of G^{ab}.
Laplacian energy of trees with at most 10 vertices
Rehman, Masood Ur ( University of Science and Technology of China (USTC), School of Mathematical Sciences. ) , Ajmal, Muhammad ( University of Science and Technology of China (USTC), School of Mathematical Sciences. ) , Kamran, Tayyab ( Department of Mathematics, QuaidIAzam University, IslamabadPakistan )
Indonesian Journal of Combinatorics Vol 2, No 1 (2018)
Publisher : Indonesian Combinatorial Society (InaCombS)
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DOI: 10.19184/ijc.2018.2.1.3
Let Tn be the set of all trees with n ≤ 10 vertices. We show that the Laplacian energy of any tree Tn is strictly between the Laplacian energy of the path Pn and the star Sn, the authors partially proving that the conjecture hold for any tree Tn, where n ≤ 10.
On star coloring of Mycielskians
Kaliraj, K. ( University of Madras ) , Kowsalya, V. ( Bharathiar University ) , Vivin, Vernold ( University College of Engineering Nagercoil )
Indonesian Journal of Combinatorics Vol 2, No 2 (2018)
Publisher : Indonesian Combinatorial Society (InaCombS)
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DOI: 10.19184/ijc.2018.2.2.3
In a search for trianglefree graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph μ(G), we now call the Mycielskian of G, which has the same clique number as G and whose chromatic number equals χ(G) + 1. In this paper, we find the star chromatic number for the Mycielskian graph of complete graphs, paths, cycles and complete bipartite graphs.
On rdynamic coloring of some graph operations
Agustin, Ika Hesti ( Mathematics Department  University of Jember
CGANT Research Group University of Jember ) , Dafik, D. ( Mathematics Education Department  University of Jember
CGANT Research Group University of Jember ) , Harsya, A. Y. ( Mathematics Department  University of Jember )
Indonesian Journal of Combinatorics Vol 1, No 1 (2016)
Publisher : Indonesian Combinatorial Society (InaCombS)
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DOI: 10.19184/ijc.2016.1.1.3
Let $G$ be a simple, connected and undirected graph. Let $r,k$ be natural number. By a proper $k$coloring of a graph $G$, we mean a map $ c : V (G) \rightarrow S$, where $S = k$, such that any two adjacent vertices receive different colors. An $r$dynamic $k$coloring is a proper $k$coloring $c$ of $G$ such that $c(N (v)) \geq min\{r, d(v)\}$ for each vertex $v$ in $V(G)$, where $N (v)$ is the neighborhood of $v$ and $c(S) = \{c(v) : v \in S\}$ for a vertex subset $S$ . The $r$dynamic chromatic number, written as $\chi_r(G)$, is the minimum $k$ such that $G$ has an $r$dynamic $k$coloring. Note that the $1$dynamic chromatic number of graph is equal to its chromatic number, denoted by $\chi(G)$, and the $2$dynamic chromatic number of graph has been studied under the name a dynamic chromatic number, denoted by $\chi_d(G)$. By simple observation it is easy to see that $\chi_r(G)\le \chi_{r+1}(G)$, however $\chi_{r+1}(G)\chi_r(G)$ can be arbitrarily large, for example $\chi(Petersen)=2, \chi_d(Petersen)=3$, but $\chi_3(Petersen)=10$. Thus, finding an exact values of $\chi_r(G)$ is significantly useful. In this paper, we will show some exact values of $\chi_r(G)$ when $G$ is an operation of special graphs.
A strict upper bound for size multipartite Ramsey numbers of paths versus stars
Jayawardene, Chula ( University of Colombo ) , Samarasekara, Lilanthi ( University of Colombo )
Indonesian Journal of Combinatorics Vol 1, No 2 (2017)
Publisher : Indonesian Combinatorial Society (InaCombS)
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DOI: 10.19184/ijc.2017.1.2.2
Let $P_n$ represent the path of size $n$. Let $K_{1,m1}$ represent a star of size $m$ and be denoted by $S_{m}$. Given a two coloring of the edges of a complete graph $K_{j \times s}$ we say that $K_{j \times s}\rightarrow (P_n,S_{m+1})$ if there is a copy of $P_n$ in the first color or a copy of $S_{m+1}$ in the second color. The size Ramsey multipartite number $m_j(P_n, S_{m+1})$ is the smallest natural number $s$ such that $K_{j \times s}\rightarrow (P_n,S_{m+1})$. Given $j,n,m$ if $s=\left\lceil \dfrac{n+m1k}{j1} \right\rceil$, in this paper, we show that the size Ramsey numbers $m_j(P_n,S_{m+1})$ is bounded above by $s$ for $k=\left\lceil \dfrac{n1}{j} \right\rceil$. Given $j\ge 3$ and $s$, we will obtain an infinite class $(n,m)$ that achieves this upper bound $s$. In the later part of the paper, will also investigate necessary and sufficient conditions needed for the upper bound to hold.
Local antimagic vertex coloring of unicyclic graphs
Nazula, Nuris Hisan ( University of Jember ) , Slamin, S ( University of Jember ) , Dafik, D ( University of Jember )
Indonesian Journal of Combinatorics Vol 2, No 1 (2018)
Publisher : Indonesian Combinatorial Society (InaCombS)
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DOI: 10.19184/ijc.2018.2.1.4
The local antimagic labeling on a graph G with ∣V∣ vertices and ∣E∣ edges is defined to be an assignment f : E → {1, 2, ⋯, ∣E∣} so that the weights of any two adjacent vertices u and v are distinct, that is, w(u) ≠ w(v) where w(u) = Σe ∈ E(u)f(e) and E(u) is the set of edges incident to u. Therefore, any local antimagic labeling induces a proper vertex coloring of G where the vertex u is assigned the color w(u). The local antimagic chromatic number, denoted by χla(G), is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper, we present the local antimagic chromatic number of unicyclic graphs that is the graphs containing exactly one cycle such as kite and cycle with two neighbour pendants.
On the local metric dimension of tfold wheel, Pn o Km, and generalized fan
Solekhah, Rokhana Ayu ( Universitas Sebelas Maret, Surakarta ) , Kusmayadi, Tri Atmojo ( Universitas Sebelas Maret, Surakarta )
Indonesian Journal of Combinatorics Vol 2, No 2 (2018)
Publisher : Indonesian Combinatorial Society (InaCombS)
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DOI: 10.19184/ijc.2018.2.2.4
Let G be a connected graph and let u, v ∈ V(G). For an ordered set W = {w1, w2, ..., wn} of n distinct vertices in G, the representation of a vertex v of G with respect to W is the nvector r(v∣W) = (d(v, w1), d(v, w2), ..., d(v, wn)), where d(v, wi) is the distance between v and wi for 1 ≤ i ≤ n. The set W is a local metric set of G if r(u ∣ W) ≠ r(v ∣ W) for every pair u, v of adjacent vertices of G. The local metric set of G with minimum cardinality is called a local metric basis for G and its cardinality is called a local metric dimension, denoted by lmd(G). In this paper we determine the local metric dimension of a tfold wheel graph, Pn ⊙ Km graph, and generalized fan graph.
Orthogonal labeling
Immanuel, Bernard ( Universitas Indonesia ) , Sugeng, Kiki A. ( Universitas Indonesia )
Indonesian Journal of Combinatorics Vol 1, No 1 (2016)
Publisher : Indonesian Combinatorial Society (InaCombS)
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DOI: 10.19184/ijc.2016.1.1.1
Let ∆G be the maximum degree of a simple connected graph G(V,E). An injective mapping P : V → R∆G is said to be an orthogonal labeling of G if uv,uw ∈ E implying (P(v) − P(u)) · (P(w) − P(u)) = 0, where · is the usual dot product defined in Euclidean space. A graph G which has an orthogonal labeling is called an orthogonal graph. This labeling is motivated by the existence of several labelings defined by some algebraic structure, i.e. harmonious labeling and group distance magic labeling. In this paper we study some preliminary results on orthogonal labeling. One of the early result is the fact that cycle graph with even vertices are orthogonal, while ones with odd vertices are not. The main results in this paper state that any graph containing K3 as its subgraph is nonorthogonal and that a graph G′ obtained from adding a pendant to a vertex in orthogonal graph G is orthogonal. In the end of the paper we state the corollary that any tree is orthogonal.