Adjacent-vertex-distinguishing-total coloring


In graph theory, a total coloring is a coloring on the vertices and edges of a graph such that:
. no adjacent vertices have the same color;
. no adjacent edges have the same color; and
. no edge and its endvertices are assigned the same color.
In 2005, Zhang et al. added a restriction to the definition of total coloring and proposed a new type of coloring defined as follows.
Let G = be a simple graph endowed with a total coloring φ, and let u be a vertex of G. The set of colors that occurs in the vertex u is defined as C = ∪. Two vertices u,v ∈ V are distinguishable if their color-sets are distinct, i.e., C ≠ C.
In graph theory, a total coloring is an adjacent-vertex-distinguishing-total-coloring if it has the following additional property:
. for every two adjacent vertices u,v of a graph G, their colors-sets are distinct from each other, i.e., C ≠ C.
The adjacent-vertex-distinguishing-total-chromatic number χat of a graph G is the least number of colors needed in an AVD-total-coloring of G.
The following lower bound for the AVD-total chromatic number can be obtained from the definition of AVD-total-coloring: If a simple graph G has two adjacent vertices of maximum degree, then χat ≥ Δ + 2. Otherwise, if a simple graph G does not have two adjacent vertices of maximum degree, then χat ≥ Δ + 1.
In 2005, Zhang et al. determined the AVD-total-chromatic number for some classes of graphs, and based in their results they conjectured the following.
AVD-Total-Coloring Conjecture.
The AVD-Total-Coloring Conjecture is known to hold for some classes of graphs, such as complete graphs, graphs with Δ=3, and all bipartite graphs.
In 2012, Huang et al. showed that χat ≤ 2Δ
for any simple graph G with maximum degree Δ > 2.
In 2014, Papaioannou and Raftopoulou described an algorithmic procedure that gives a
7-AVD-total-colouring for any 4-regular graph.