Adjacent Vertex Distinguishing Acyclic Edge Coloring of the Cartesian Product of Graphs
نویسندگان
چکیده
Let G be a graph and χaa(G) denotes the minimum number of colors required for an acyclic edge coloring of G in which no two adjacent vertices are incident to edges colored with the same set of colors. We prove a general bound for χaa(G□H) for any two graphs G and H. We also determine exact value of this parameter for the Cartesian product of two paths, Cartesian product of a path and a cycle, Cartesian product of two trees, hypercubes. We show that χaa(Cm□Cn) is at most 6 fo every m ≥ 3 and n ≥ 3. Moreover in some cases we find the exact value of χaa(Cm□Cn).
منابع مشابه
Some results on vertex-edge Wiener polynomials and indices of graphs
The vertex-edge Wiener polynomials of a simple connected graph are defined based on the distances between vertices and edges of that graph. The first derivative of these polynomials at one are called the vertex-edge Wiener indices. In this paper, we express some basic properties of the first and second vertex-edge Wiener polynomials of simple connected graphs and compare the first and second ve...
متن کاملEdge-coloring Vertex-weightings of Graphs
Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $n$. A $k$-vertex weightings of a graph $G$ is a mapping $w: V(G) to {1, ldots, k}$. A $k$-vertex weighting induces an edge labeling $f_w: E(G) to N$ such that $f_w(uv)=w(u)+w(v)$. Such a labeling is called an {it edge-coloring k-vertex weightings} if $f_{w}(e)not= f_{w}(echr(chr(chr('39')39chr('39'))39chr(chr('39')39chr('39'...
متن کاملOn Acyclic Vertex Coloring of Grid like graphs
d-dimensional partial tori are graphs that can be expressed as cartesian product of d graphs each of which is an induced path or cycle. Some well known graphs like d-dimensional hypercubes, meshes and tori are examples belong to this class. Muthu et al.[MNS06] have studied the problem of acyclic edge coloring for such graphs. We try to explore the acyclic vertex coloring problem for these graph...
متن کاملGame coloring the Cartesian product of graphs
This article proves the following result: Let G and G′ be graphs of orders n and n′, respectively. Let G∗ be obtained from G by adding to each vertex a set of n′ degree 1 neighbors. If G∗ has game coloring number m and G′ has acyclic chromatic number k, then the Cartesian product G G′ has game chromatic number at most k(k+m − 1). As a consequence, the Cartesian product of two forests has game c...
متن کاملAdjacent Vertex Distinguishing Edge-Colorings
An adjacent vertex distinguishing edge-coloring of a simple graph G is a proper edge-coloring of G such that no pair of adjacent vertices meets the same set of colors. The minimum number of colors χa(G) required to give G an adjacent vertex distinguishing coloring is studied for graphs with no isolated edge. We prove χa(G) ≤ 5 for such graphs with maximum degree Δ(G) = 3 and prove χa(G) ≤ Δ(G) ...
متن کامل