Indonesian Journal of Combinatorics最新文献

Let H and G be two simple graphs. The concept of an H-magic 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 H-magic 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 K_n, 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.

Let G be a connected graph and let u, v ∈ V(G). For an ordered set W = {w₁, w₂, ..., w_n} of n distinct vertices in G, the representation of a vertex v of G with respect to W is the n-vector r(v∣W) = (d(v, w₁), d(v, w₂), ..., d(v, w_n)), where d(v, w_i) is the distance between v and w_i 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 t-fold wheel graph, P_n ⊙ K_m graph, and generalized fan graph.

A tree T(V, E) is graceful if there exists an injective function f from the vertex set V(T) into the set {0, 1, 2, ..., ∣V∣ − 1} which induces a bijective function fʹ from the edge set E(T) onto the set {1, 2, ..., ∣E∣}, with fʹ(uv) = ∣f(u) − f(v)∣ for every edge {u, v} ∈ E. 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.

In a search for triangle-free 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.

A Γ-supermagic labeling of a graph G = (V, E) with ∣E∣ = k is a bijection from E to an Abelian group Γ of order k such that the sum of labels of all incident edges of every vertex x ∈ V is equal to the same element μ ∈ Γ. We present a Z_2nm-supermagic labeling of Cartesian product of two cycles, C_n□C_m for n odd. This along with an earlier result by Ivančo proves that a Z_2nm-supermagic labeling of C_n□C_m exists for every n, m ≥ 3.

An edge magic total (EMT) labeling of a graph G = (V, E) is a bijection from the set of vertices and edges to a set of numbers defined by λ : V ∪ E → {1, 2, ..., ∣V∣ + ∣E∣} with the property that for every xy ∈ E, the weight of xy equals to a constant k, that is, λ(x) + λ(y) + λ(xy) = k for some integer k. In this paper given the construction of an EMT labeling for certain lexicographic product $C_{4(2r+1)}circ overline{K_2}$, cycle with chords ^[c]tC_n, unions of paths mP_n, and unions of cycles and paths m(C_{n1(2r + 1)} ∪ (2r + 1)P_n2).