vertex equitable labelings of transformed trees

Vertex Equitable Labelings of Transformed Trees

4−equitable Tree Labelings

A tree is a vertex-edge graph that is connected and contains no cycles. A 4−equitable labeling of a graph is an assignment of labels {0, 1, 2, 3} to the vertices. The edge labels are the absolute difference of the labels of the vertices that they are incident to. The labels must be distributed as evenly as possible amongst the vertices and they must also be distributed as evenly as possible amo...

Vertex-magic labelings: mutations

In this paper, we are studying vertex-magic total labelings of simple graphs. We introduce a procedure called mutation which transforms one labeling into another by swapping sets of edges among vertices. The result of a mutation may be a different labeling of the same graph or a labeling of a different graph. Mutation proves to be a remarkably fruitful process— for example we are able to genera...

Equitable vertex arboricity of graphs

An equitable (t, k, d)-tree-coloring of a graph G is a coloring to vertices of G such that the sizes of any two color classes differ by at most one and the subgraph induced by each color class is a forest of maximum degree at most k and diameter at most d. The minimum t such that G has an equitable (t′, k, d)-tree-coloring for every t′ ≥ t is called the strong equitable (k, d)-vertex-arboricity...

Vertex-antimagic total labelings of graphs

In this paper we introduce a new type of graph labeling, the (a, d)vertex-antimagic total labeling, which is a generalization of several other types of labelings. A connected graph G(V, E) is said to be (a, d)-vertex-antimagic total if there exist positive integers a, d and a bijection λ : V ∪ E → {1, 2, . . . , |V | + |E|} such that the induced mapping gλ : V → W is also a bijection, where W =...

Vertex Equitable Labeling of Double Alternate Snake Graphs

Let G be a graph with p vertices and q edges and A = {0, 1, 2, . . . , [q/2]}. A vertex labeling f : V (G) → A induces an edge labeling f∗ defined by f∗(uv) = f(u) + f(v) for all edges uv. For a ∈ A, let vf (a) be the number of vertices v with f(v) = a. A graph G is said to be vertex equitable if there exists a vertex labeling f such that for all a and b in A, |vf (a) − vf (b)| ≤ 1 and the indu...