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All Journal Electronic Journal of Graph Theory and Applications (EJGTA) Journal of the Indonesian Mathematical Society Jurnal Matematika & Sains Journal of Mathematical and Fundamental Sciences Indonesian Journal of Combinatorics
Edy Tri Baskoro
Institut Teknologi Bandung
Astronomy Chemistry Computer Science & IT Decision Sciences, Operations Research & Management Earth & Planetary Sciences Electrical & Electronics Engineering Mathematics Physics

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Trees with Certain Locating-chromatic Number Dian Kastika Syofyan; Edy Tri Baskoro; Hilda Assiyatun
Journal of Mathematical and Fundamental Sciences Vol. 48 No. 1 (2016)
Publisher : Institute for Research and Community Services (LPPM) ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/j.math.fund.sci.2016.48.1.4

Abstract

The locating-chromatic number of a graph G can be defined as the cardinality of a minimum resolving partition of the vertex set V(G) such that all vertices have distinct coordinates with respect to this partition and every two adjacent vertices in G are not contained in the same partition class. In this case, the coordinate of a vertex v in G is expressed in terms of the distances of v to all partition classes. This concept is a special case of the graph partition dimension notion. Previous authors have characterized all graphs of order n with locating-chromatic number either n or n-1. They also proved that there exists a tree of order n, n≥5, having locating-chromatic number k if and only if k âˆˆ{3,4,"¦,n-2,n}. In this paper, we characterize all trees of order n with locating-chromatic number n - t, for any integers n and t, where n > t+3 and 2 ≤ t < n/2.
On The Partition Dimension of Disconnected Graphs Debi Oktia Haryeni; Edy Tri Baskoro; Suhadi Wido Saputro
Journal of Mathematical and Fundamental Sciences Vol. 49 No. 1 (2017)
Publisher : Institute for Research and Community Services (LPPM) ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/j.math.fund.sci.2017.1.2

Abstract

For a graph G=(V,E), a partition Ω={O1,O2,"¦,Ok} of the vertex set V is called a resolving partition if every pair of vertices u,v ∈ V(G) have distinct representations under Ω. The partition dimension of G is the minimum integer k such that G has a resolving k-partition. Many results in determining the partition dimension of graphs have been obtained. However, the known results are limited to connected graphs. In this study, the notion of the partition dimension of a graph is extended so that it can be applied to disconnected graphs as well. Some lower and upper bounds for the partition dimension of a disconnected graph are determined (if they are finite). In this paper, also the partition dimensions for some classes of disconnected graphs are given.
Restricted Size Ramsey Number Involving Matching and Graph of Order Five Denny Riama Silaban; Edy Tri Baskoro; Saladin Uttunggadewa
Journal of Mathematical and Fundamental Sciences Vol. 52 No. 2 (2020)
Publisher : Institute for Research and Community Services (LPPM) ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/j.math.fund.sci.2020.52.2.1

Abstract

Harary and Miller (1983) started the research on the (restricted) size Ramsey number for a pair of small graphs. They obtained the values for some pairs of small graphs with order not more than four. In the same year, Faudree and Sheehan continued the research and extended the result to all pairs of small graphs with order not more than four. Moreover, in 1998, Lortz and Mengenser gave the size Ramsey number and the restricted size Ramsey number for all pairs of small forests with order not more than five. Recently, we gave the restricted size Ramsey number for a path of order three and any connected graph of order five. In this paper, we continue the research on the (restricted) size Ramsey number involving small graphs by investigating the restricted size Ramsey number for matching with two edges versus any graph of order five with no isolates.
The Uniqueness of Almost Moore Digraphs with Degree 4 And Diameter 2 Rinovia Simanjuntak; Edy Tri Baskoro
Journal of Mathematical and Fundamental Sciences Vol. 32 No. 1 (2000)
Publisher : Institute for Research and Community Services (LPPM) ITB

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Abstract

Abstract. It is well known that Moore digraphs of degree d > 1 and diameter k > 1 do not exist. For degrees 2 and 3, it has been shown that for diameter k ≥ 3 there are no almost Moore digraphs, i.e. the diregular digraphs of order one less than the Moore bound. Digraphs with order close to the Moore bound arise in the construction of optimal networks. For diameter 2, it is known that almost Moore digraphs exist for any degree because the line digraphs of complete digraphs are examples of such digraphs. However, it is not known whether these are the only almost Moore digraphs. It is shown that for degree 3, there are no almost Moore digraphs of diameter 2 other than the line digraph of K4. In this paper, we shall consider the almost Moore digraphs of diameter 2 and degree 4. We prove that there is exactly one such digraph, namely the line digraph of K5. Ketunggalan Graf Berarah Hampir Moore dengan Derajat 4 dan Diameter 2Sari. Telah lama diketahui bahwa tidak ada graf berarah Moore dengan derajat d>1 dan diameter k>1. Lebih lanjut, untuk derajat 2 dan 3, telah ditunjukkan bahwa untuk diameter t>3, tidak ada graf berarah Hampir Moore, yakni graf berarah teratur dengan orde satu lebih kecil dari batas Moore. Graf berarah dengan orde mendekati batas Moore digunakan dalam pcngkonstruksian jaringan optimal. Untuk diameter 2, diketahui bahwa graf berarah Hampir Moore ada untuk setiap derajat karena graf berarah garis (line digraph) dari graf komplit adalah salah satu contoh dari graf berarah tersebut. Akan tetapi, belum dapat dibuktikan apakah graf berarah tersebut merupakan satu-satunya contoh dari graf berarah Hampir Moore tadi. Selanjutnya telah ditunjukkan bahwa untuk derajat 3, tidak ada graf berarah Hampir Moore diameter 2 selain graf berarah garis dari K4. Pada makalah ini, kita mengkaji graf berarah Hampir Moore diameter 2 dan derajat 4. Kita buktikan bahwa ada tepat satu graf berarah tersebut, yaitu graf berarah garis dari K5.
On Size Bipartite and Tripartite Ramsey Numbers for The Star Forest and Path on 3 Vertices Anie Lusiani; Edy Tri Baskoro; Suhadi Wido Saputro
Journal of Mathematical and Fundamental Sciences Vol. 52 No. 1 (2020)
Publisher : Institute for Research and Community Services (LPPM) ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/j.math.fund.sci.2020.52.1.1

Abstract

For simple graphs G and H the size multipartite Ramsey number mj(G,H) is the smallest natural number t such that any arbitrary red-blue coloring on the edges of Kjxt contains a red G or a blue H as a subgraph. We studied the size tripartite Ramsey numbers m3(G,H) where G=mK1,n and H=P3. In this paper, we generalize this result. We determine m3(G,H) where G is a star forest, namely a disjoint union of heterogeneous stars, and H=P3. Moreover, we also determine m2(G,H) for this pair of graphs G and H.
A Note on Almost Moore Diagraphs of Degree Three Edy Tri Baskoro; Mirka Miller; Josef Siran
Journal of Mathematical and Fundamental Sciences Vol. 30 No. 1 (1998)
Publisher : Institute for Research and Community Services (LPPM) ITB

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Abstract

It is well known that Moore digraphs do not exist except for trivial cases (degree 1 or diameter 1), but there are digraphs of diameter two and arbitrary degree which miss the Moore bound by one. No examples of such digraphs of diameter at least three are known, although several necessary conditions for their existence have been obtained. A particularly interesting necessary condition for the existence of a digraph of degree three and diameter k > 3 of order one less than the Moore bound is that the number of its arcs be divisible by k + 1. In this paper we derive a new necessary condition (in terms of cycles of the so-called repeat permutation) for the existence of such digraphs of degree three. As a consequence we obtain that a digraph of degree three and diameter k ≥ 3 which misses the Moore bound by one cannot be a Cayley digraph of an Abelian group. Catatan Untuk Keberadaan Graf Berarah Hampir Moore Derajat 3Telah lama diketahui bahwa tidak ada graf berarah dengan orde (jumlah titiknya) sama dengan batas Moore, kecuali untuk kasus-kasus trivial, yakni untuk derajat 1 atau diameter 1; tetapi, ada graf berarah dengan diameter 2 untuk sebarang derajat dengan orde satu lebih kecil dari batas Moore. Hingga kini belum dapat ditunjukkan adanya contoh graf berarah yang sejenis dengan diameter paling sedikit 3, walaupun beberapa syarat perlu akan keberadaannya telah diberikan. Salah satu syarat perlu yang cukup menarik untuk keberadaan graf berarah dengan derajat 3, diameter k > atau = 3 dan orde satu lebih kecil dari batas Moore adalah bahwa jumlah busur yang dimilikinya harus dapat dibagi oleh bilangan k+1.
On the Ramsey number of 4-cycle versus wheel Enik Noviani; Edy Tri Baskoro
Indonesian Journal of Combinatorics Vol 1, No 1 (2016)
Publisher : Indonesian Combinatorial Society (InaCombS)

Show Abstract | Download Original | 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. 
Locating-chromatic number of the edge-amalgamation of trees Hilda Assiyatun; Dian Kastika Syofyan; Edy Tri Baskoro
Indonesian Journal of Combinatorics Vol 4, No 2 (2020)
Publisher : Indonesian Combinatorial Society (InaCombS)

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.19184/ijc.2020.4.2.6

Abstract

The investigation on the locating-chromatic number for graphs was initially studied by Chartrand et al. on 2002. This concept is in fact a special case of the partition dimension for graphs. Even though this topic has received much attention, the current progress is still far from satisfaction. We can define the locating-chromatic number of a graph G as the smallest integer k such that there exists a proper k-coloring on the vertex-set of G such that all vertices have distinct coordinates (color codes) with respect to this coloring. Not like the metric dimension of any tree which is completely solved, the locating-chromatic number for most types of trees are still open. In this paper, we study the locating-chromatic number of trees. In particular, we give lower and upper bounds of the locating-chromatic number of trees formed by an edge-amalgamation of the collection of smaller trees. We also show that the bounds are tight.
On size multipartite Ramsey numbers for stars Anie Lusiani; Edy Tri Baskoro; Suhadi Wido Saputro
Indonesian Journal of Combinatorics Vol 3, No 2 (2019)
Publisher : Indonesian Combinatorial Society (InaCombS)

Show Abstract | Download Original | Original Source | Check in Google Scholar | Full PDF (322.917 KB) | DOI: 10.19184/ijc.2019.3.2.4

Abstract

Burger and Vuuren defined the size multipartite Ramsey number for a pair of complete, balanced, multipartite graphs mj(Kaxb,Kcxd), for natural numbers a,b,c,d and j, where a,c >= 2, in 2004. They have also determined the necessary and sufficient conditions for the existence of size multipartite Ramsey numbers mj(Kaxb,Kcxd). Syafrizal et al. generalized this definition by removing the completeness requirement. For simple graphs G and H, they defined the size multipartite Ramsey number mj(G,H) as the smallest natural number t such that any red-blue coloring on the edges of Kjxt contains a red G or a blue H as a subgraph. In this paper, we determine the necessary and sufficient conditions for the existence of multipartite Ramsey numbers mj(G,H), where both G and H are non complete graphs. Furthermore, we determine the exact values of the size multipartite Ramsey numbers mj(K1,m, K1,n) for all integers m,n >= 1 and j = 2,3, where K1,m is a star of order m+1. In addition, we also determine the lower bound of m3(kK1,m, C3), where kK1,m is a disjoint union of k copies of a star K1,m and C3 is a cycle of order 3.
The locating-chromatic number and partition dimension of fibonacene graphs Ratih Suryaningsih; Edy Tri Baskoro
Indonesian Journal of Combinatorics Vol 3, No 2 (2019)
Publisher : Indonesian Combinatorial Society (InaCombS)

Show Abstract | Download Original | Original Source | Check in Google Scholar | Full PDF (638.687 KB) | DOI: 10.19184/ijc.2019.3.2.5

Abstract

Fibonacenes are unbranched catacondensed benzenoid hydrocarbons in which all the non-terminal hexagons are angularly annelated. A hexagon is said to be angularly annelated if the hexagon is adjacent to exactly two other hexagons and possesses two adjacent vertices of degree 2. Fibonacenes possess remarkable properties related with Fibonacci numbers. Various graph properties of fibonacenes have been extensively studied, such as their saturation numbers, independence numbers and Wiener indices. In this paper, we show that the locating-chromatic number of any fibonacene graph is 4 and the partition dimension of such a graph is 3.
Co-Authors A Asmiati Ahmad Erfanian Anisa Rarasati Arfin Arfin Bana Kartasasmita Bana Kartasasmita Brilly Maxel Salindeho BUDI RAHADJENG Chula Janak Jayawardene Debi Oktia Haryeni Debi Oktia Haryeni Debi Oktia Haryeni Denny Riama Silaban Denny Riama Silaban Dian Kastika Syofyan Dian Kastika Syofyan Dinny Fitriani Djoko Suprijanto Enik Noviani Erfanian, Ahmad Faisal Susanto Gholamreza Abrishami Gregory Benedict Tanidi Hilal A Ganie Hilda Assiyatun Hilda Assiyatun Hilda Assiyatun Hilda Assiyatun Hilda Assiyatun Hilda Assiyatun Hilda Assiyatun Hilda Assiyatun, Hilda I Made Arnawa I Made Arnawa I Wayan Palton Anuwiksa Josef Siran Korivand, Meysam Kristiana Wijaya Lilanthi Samarasekara Lusiani, Anie Maya Nabila Maya Nabila Meysam Korivand Mirka Miller Mojdeh, Doost Ali Mostafa Tavakoli Muhammad Kamran Siddiqui Muhammad Ridwan Muhammad Salman Alfarisi Nawawi Mushtaq Shah N. Nurdin Nobuaki Obata Nurdin Nurdin Ratih Suryaningsih Rika Yanti Rinovia Simanjuntak Rinovia Simanjuntak Rinovia Simanjuntak Rinovia Simanjuntak Rinovia Simanjuntak Saladin Uttunggadewa Saladin Uttunggadewa Saladin Uttunggadewa Salindeho, Brilly Maxel Shariefuddin Pirzada Shariefuddin Pirzada Suhadi Wido Saputro Suhadi Wido Saputro Suhadi Wido Saputro Suhadi Wido Saputro Suhadi Wido Saputro Suhadi Wido Saputro Susilawati, S. Syafrizal Sy Utari sumarno Utari sumarno