Indonesian Journal of Combinatorics
ISSN : 25412205     EISSN : -
Indonesian Journal of Combinatorics (IJC) publishes current research articles in any area of combinatorics and graph theory such as graph labelings, optimal network problems, metric dimension, graph coloring, rainbow connection and other related topics. IJC is published by the Indonesian Combinatorial Society (InaCombS), CGANT Research Group Universitas Jember (UNEJ), and Department of Mathematics Universitas Indonesia (UI).
Articles 5 Documents
Search results for , issue " Vol 1, No 1 (2016)" : 5 Documents clear
On the Ramsey number of 4-cycle versus wheel Noviani, Enik; Baskoro, Edy Tri
Indonesian Journal of Combinatorics Vol 1, No 1 (2016)
Publisher : Indonesian Combinatorial Society (InaCombS)

Show Abstract | Original Source | Check in Google Scholar | Full PDF (529.961 KB) | DOI: 10.19184/ijc.2016.1.1.2

Abstract

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. 
On r-dynamic coloring of some graph operations Agustin, Ika Hesti; Dafik, D.; Harsya, A. Y.
Indonesian Journal of Combinatorics Vol 1, No 1 (2016)
Publisher : Indonesian Combinatorial Society (InaCombS)

Show Abstract | Original Source | Check in Google Scholar | Full PDF (190.124 KB) | DOI: 10.19184/ijc.2016.1.1.3

Abstract

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.
Orthogonal labeling Immanuel, Bernard; Sugeng, Kiki A.
Indonesian Journal of Combinatorics Vol 1, No 1 (2016)
Publisher : Indonesian Combinatorial Society (InaCombS)

Show Abstract | Original Source | Check in Google Scholar | Full PDF (218.824 KB) | DOI: 10.19184/ijc.2016.1.1.1

Abstract

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 non-orthogonal 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.
Size multipartite Ramsey numbers for small paths versus books Jayawardene, Chula J.; Ratnayake, Jayampathy
Indonesian Journal of Combinatorics Vol 1, No 1 (2016)
Publisher : Indonesian Combinatorial Society (InaCombS)

Show Abstract | Original Source | Check in Google Scholar | Full PDF (209.036 KB) | DOI: 10.19184/ijc.2016.1.1.4

Abstract

Given $j \ge 2$,  for  graphs $G$ and $H$, the size Ramsey multipartite number $m_j(G, H)$ is defined as the smallest natural number $t$ such that any blue red coloring of the edges of the  graph $K_{j \times t}$, necessarily containes a red $G$ or a blue $H$ as subgraphs. Let the book with $n$ pages is defined as the graph $K_1 + K_{1,n}$ and denoted by $B_n$. In this paper, we  obtain the exact values of the size Ramsey numbers $m_j(P_3, H)$ for $j \ge 3$ where  $H$ is a book $B_n$. We also derive some upper and lower bounds for the size Ramsey numbers $m_j(P_4, H)$ where  $H$ is a book $B_n$.
Dominating number of distance two of corona products of graphs Umilasari, Reni; Darmaji, Darmaji
Indonesian Journal of Combinatorics Vol 1, No 1 (2016)
Publisher : Indonesian Combinatorial Society (InaCombS)

Show Abstract | Original Source | Check in Google Scholar | Full PDF (180.088 KB) | DOI: 10.19184/ijc.2016.1.1.5

Abstract

Dominating set $S$ in graph $G=(V,E)$ is a subset of $V(G)$ such that every vertex of $G$ which is not element of $S$ are connected and have distance one to $S$. Minimum cardinality among dominating sets in a graph $G$ is called dominating number of graph $G$ and denoted by $\gamma(G)$. While dominating set ofdistance two which denoted by $S_2$ is a subset of $V(G)$ such that every vertex of $G$ which is not element of $S$ are connected and have maximum distance two to $S_2$. Dominating number of distance two $\gamma_2(G)$ is minimum cardinality of dominating set of distance two $S_2$. The corona $G \odot H$ of two graphs $G$ and $H$ where $G$ has $p$ vertices and $q$ edges is defined as the graph G obtained by taking one copy of $G$ and $p$ copies of $H$, and then joining by an edge the $i-th$ vertex of $G$ to every vertex in the $i-th$ copy of $H$. In this paper, we determine the dominating number of distance two of paths and cycles. We also determine the dominating number of distance two of corona product of path and any graphs as well as cycle and any graphs.

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