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- Adjacent-vertex-distinguishing-total_coloring abstract "In graph theory, a total coloring is a coloring on the vertices and edges of a graph such that:(1). no adjacent vertices have the same color;(2). no adjacent edges have the same color; and(3). 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 = (V,E) 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(u) = {φ(u)} ∪ {φ(uv) | uv ∈ E(G)}. Two vertices u,v ∈ V(G) are distinguishable if their color-sets are distinct, i.e., C(u) ≠ C(v).In graph theory, a total coloring is an adjacent-vertex-distinguishing-total-coloring (AVD-total-coloring) if it has the following additional property:(4). for every two adjacent vertices u,v of a graph G, their colors-sets are distinct from each other, i.e., C(u) ≠ C(v). The adjacent-vertex-distinguishing-total-chromatic number χat(G) 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(G) ≥ Δ(G) + 2. Otherwise, if a simple graph G does not have two adjacent vertices of maximum degree, then χat(G) ≥ Δ(G) + 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. (Zhang et al.)χat(G) ≤ Δ(G) + 3.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.".
- Adjacent-vertex-distinguishing-total_coloring thumbnail Avd-total-coloring-of-complete-graph-K4.svg?width=300.
- Adjacent-vertex-distinguishing-total_coloring wikiPageExternalLink 12_39_mc_rev_vagner_ctfgi.pdf.
- Adjacent-vertex-distinguishing-total_coloring wikiPageID "39479572".
- Adjacent-vertex-distinguishing-total_coloring wikiPageRevisionID "575688035".
- Adjacent-vertex-distinguishing-total_coloring subject Category:Graph_coloring.
- Adjacent-vertex-distinguishing-total_coloring subject Category:Graph_theory.
- Adjacent-vertex-distinguishing-total_coloring comment "In graph theory, a total coloring is a coloring on the vertices and edges of a graph such that:(1). no adjacent vertices have the same color;(2). no adjacent edges have the same color; and(3). 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 = (V,E) be a simple graph endowed with a total coloring φ, and let u be a vertex of G.".
- Adjacent-vertex-distinguishing-total_coloring label "Adjacent-vertex-distinguishing-total coloring".
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- Adjacent-vertex-distinguishing-total_coloring sameAs Q17005461.
- Adjacent-vertex-distinguishing-total_coloring sameAs Q17005461.
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- Adjacent-vertex-distinguishing-total_coloring depiction Avd-total-coloring-of-complete-graph-K4.svg.
- Adjacent-vertex-distinguishing-total_coloring isPrimaryTopicOf Adjacent-vertex-distinguishing-total_coloring.