Edy Tri Baskoro

On the total vertex irregularity strength of trees

A vertex irregular total k-labelling @l:V(G)@?E(G)@?{1,2,...,k} of a graph G is a labelling of vertices and edges of G done in such a way that for any different vertices x and y, their weights wt(x) and wt(y) are distinct. The weight wt(x) of a vertex x is the sum of the label of x and the labels of all edges incident with x. The minimum k for which a graph G has a vertex irregular total k-labelling is called the total vertex irregularity strength of G, denoted by tvs(G). In this paper, we determine the total vertex irregularity strength of trees.

The metric dimension of the lexicographic product of graphs

Abstract A set of vertices W resolves a graph G if every vertex is uniquely determined by its coordinate of distances to the vertices in W . The minimum cardinality of a resolving set of G is called the metric dimension of G . In this paper, we consider a graph which is obtained by the lexicographic product between two graphs. The lexicographic product of graphs G and H , which is denoted by G ∘ H , is the graph with vertex set V ( G ) × V ( H ) = { ( a , v ) | a ∈ V ( G ) , v ∈ V ( H ) } , where ( a , v ) is adjacent to ( b , w ) whenever a b ∈ E ( G ) , or a = b and v w ∈ E ( H ) . We give the general bounds of the metric dimension of a lexicographic product of any connected graph G and an arbitrary graph H . We also show that the bounds are sharp.

On the Structure of Digraphs with Order Close to the Moore Bound

Abstract. The Moore bound for a diregular digraph of degree E5>, k and diameter k is . It is known that digraphs of order do not exist for d>1 and k>1 ([24] or [6]). In this paper we study digraphs of degree E5>, k, diameter k and order , denoted by (d, k)-digraphs. Miller and Fris showed that (2, k)-digraphs do not exist for k≥3 [22]. Subsequently, we gave a necessary condition of the existence of (3, k)-digraphs, namely, (3, k)-digraphs do not exist if k is odd or if k+1 does not divide [3]. The (E5>, k, 2)-digraphs were considered in [4]. In this paper, we present further necessary conditions for the existence of (d, k)-digraphs. In particular, for , we show that a (d, k)-digraph contains either no cycle of length k or exactly one cycle of length k.

Note: The Ramsey numbers of large cycles versus wheels

In this paper we show that the Ramsey number R(Cn ,W m) = 2n − 1 for even m and n 5m/2 − 1.

Combinatorial Geometry and Graph Theory

This volume consists of the refereed papers presented at the Indonesia-Japan Joint Conference on Combinatorial Geometry and Graph Theory (IJCCGGT 2003), held on September 13 16, 2003 at ITB, Bandung, Indonesia. This conf- ence can also be considered as a series of the Japan Conference on Discrete and Computational Geometry (JCDCG), which has been held annually since 1997. The ?rst ?ve conferences of the series were held in Tokyo, Japan, the sixth in Manila, the Philippines, in 2001, and the seventh in Tokyo, Japan in 2002. The proceedings of JCDCG 1998, JCDCG 2000 and JCDCG 2002 were p- lished by Springer as part of the series Lecture Notes in Computer Science: LNCS volumes 1763, 2098 and 2866, respectively. The proceedings of JCDCG 2001 were also published by Springer as a special issue of the journal Graphs and Combinatorics, Vol. 18, No. 4, 2002. TheorganizersaregratefultotheDepartmentofMathematics,InstitutTek- logi Bandung (ITB) and Tokai University for sponsoring the conference. We also thank all program committee members and referees for their excellent work. Our big thanks to the principal speakers: Hajo Broersma, Mikio Kano, Janos Pach andJorgeUrrutia.Finally,ourthanksalsogoestoallourcolleagueswhoworked hard to make the conference enjoyable and successful. August 2004 Jin Akiyama Edy Tri Baskoro Mikio Kano Organization The Indonesia-Japan Joint Conference on Combinatorial Geometry and Graph Theory (IJCCGGT) 2003 was organized by the Department of Mathematics, InstitutTeknologiBandung(ITB)IndonesiaandRIED,TokaiUniversity,Japan

H-magic covering on some classes of graphs

For a graph G(V,E), an edge-covering of G is a family of different subgraphs H1,…Hk such that any edge of E belongs to at least one of the subgraphs Hi, 1 ≤ i ≤ k. If every Hi is isomorphic to a given graph H, then G admits an H-covering. Graph G is said to be H-magic if G has an H-covering and there is a total labeling f:V∪E→{1,2,…|V|+|E|} such that for each subgraph H′ = (V′,E′) of G isomorphic to H,ΣνϵV′f(ν)+ΣeϵE′f(e) is fixed constant. Furthermore, if f(V) = }1,2,…|V|{ then G is called H-supermagic. The sum of all vertex and edge labels on H, under a labeling f, is denoted by Σf(H). In this paper we study H-supermagic labeling for some classes of trees such as a double star, a caterpillar, a firecracker and a banana tree.

On Ramsey Numbers for Trees Versus Wheels of Five or Six Vertices

Abstract. For given two graphs G dan H, the Ramsey numberR(G,H) is the smallest positive integer n such that every graph F of order n must contain G or the complement of F must contain H. In [12], the Ramsey numbers for the combination between a star Sn and a wheel Wm for m=4,5 were shown, namely, R(Sn,W4)=2n−1 for odd n and n≥3, otherwise R(Sn,W4)=2n+1, and R(Sn,W5)=3n−2 for n≥3. In this paper, we shall study the Ramsey number R(G,Wm) for G any tree Tn. We show that if Tn is not a star then the Ramsey number R(Tn,W4)=2n−1 for n≥4 and R(Tn,W5)=3n−2 for n≥3. We also list some open problems.

Digraphs of degree 3 and order close to the Moore bound

It is known tht Moore digraphs of degree d > 1 and diameter k > 1 do not exist (see [20] or [5]). Furthermore, for degree 2, it is shown tht for l ≥ 3 there are no digraphs of order “close” to, i.e., one less than Moore bound [18]. In this paper, we shall consider digraphs of diameter k, degree 3 and number of vertices one less than Moore bound. We give a necessary condition for the existence of such digraphs and, using this condition, we deduce that such digraphs do not exist for infinitely many values of the diameter.

On energy, Laplacian energy and

For a graph

p

G