Vertex-distinguishing Total Coloring Research Articles

Adjacent vertex distinguishing total coloring in split graphs

An adjacent vertex distinguishing (AVD-) total coloring of a graph G is a total coloring such that any two adjacent vertices u and v have distinct sets of colors, that is, C(u)≠C(v), where C(v) is the set of colors of the edges incident to v and the color of v. The adjacent vertex distinguishing (AVD)-total chromatic number of a graph G, χa″(G) is the minimum integer k such that there exists an AVD-total coloring of G using k colors. It is known that χa″(G)≥Δ+1, where Δ is the maximum degree of the graph. The AVD-total coloring conjecture states that for any graph G, χa″(G)≤Δ+3. In this paper, we study AVD-total coloring in split graphs. We verify the AVD-total coloring conjecture for split graphs and classify certain classes of split graphs according to their AVD-total chromatic number.

Strong Vertex-distinguishing Total Coloring Algorithm for Complete Graphs based on Equitable Coloring

Strong Vertex-distinguishing Total Coloring Algorithm for Complete Graphs based on Equitable Coloring

Far East Journal of Mathematical Sciences (FJMS) Volume 118, Issue 1, Pages 77 - 105 (September 2019) http://dx.doi.org/10.17654/MS118010077 ADJACENT VERTEX DISTINGUISHING TOTAL COLORING OF TENSOR PRODUCT OF GRAPHS

Far East Journal of Mathematical Sciences (FJMS) Volume 118, Issue 1, Pages 77 - 105 (September 2019) http://dx.doi.org/10.17654/MS118010077 ADJACENT VERTEX DISTINGUISHING TOTAL COLORING OF TENSOR PRODUCT OF GRAPHS

Smarandachely Adjacent Vertex Distinguishing Edge Coloring of Some Graphs

A Series of new coloring problems (such as vertex distinguishing edge coloring, vertex distinguishing total coloring, adjacent vertex distinguishing edge coloring, smarandachely adjacent vertex distinguishing edge coloring)arise with the divelopment of computer science and information science. At present, there are relatively few articles on smarandachely adjacent vertex distinguishing edge coloring. This paper was to solve the problem of the smarandachely adjacent vertex distinguishing edge coloring of graph C2m×Cn.

2-distance vertex-distinguishing total coloring of graphs

A [Formula: see text]-distance vertex-distinguishing total coloring of a graph [Formula: see text] is a proper total coloring of [Formula: see text] such that any pair of vertices at distance [Formula: see text] have distinct sets of colors. The [Formula: see text]-distance vertex-distinguishing total chromatic number [Formula: see text] of [Formula: see text] is the minimum number of colors needed for a [Formula: see text]-distance vertex-distinguishing total coloring of [Formula: see text]. In this paper, we determine the [Formula: see text]-distance vertex-distinguishing total chromatic number of some graphs such as paths, cycles, wheels, trees, unicycle graphs, [Formula: see text], and [Formula: see text]. We conjecture that every simple graph [Formula: see text] with maximum degree [Formula: see text] satisfies [Formula: see text].

On the Vertex-Distinguishing Total Coloring of 𝑷𝒏 ∨ 𝑷𝒏 and 𝑪𝒏 ∨ 𝑪

Let 𝐺(𝑉, 𝐸) be a simple graph, 𝑓 is a mapping from 𝑉(𝐺) ∪ 𝐸(𝐺) to {1,2, ⋯ , 𝑘}. Let 𝐶𝑓 (𝑣) = {𝑓(𝑣)} ∪ {𝑓(𝑣𝑤)|𝑤 ∈ 𝑉(𝐺), 𝑣𝑤 ∈ 𝐸(𝐺)} for every 𝑣 ∈ 𝐸(𝐺) . If 𝑓 is a K-proper-total-coloring, and for ∀𝑢, 𝑣 ∈ 𝑉(𝐺) , we have 𝐶𝑓 (𝑢) ≠𝐶𝑓 (𝑣) , then 𝑓 is called the k-vertex-distinguishing total coloring (k-VDTC for shot). Let 𝜒𝑣𝑡 ′ (𝐺) = min {𝑘|𝐺 has a k-vertex-distinguishing total colorint}. Then 𝜒𝑣𝑡 ′ (𝐺) is called the vertex-distinguishing total chromatic number. The total chromatic number on 𝑃𝑛 ∨ 𝑃𝑛 and 𝐶𝑛⋁𝐶𝑛.

General Vertex-Distinguishing Total Coloring of Graphs

The general vertex-distinguishing total chromatic number of a graphGis the minimum integerk, for which the vertices and edges ofGare colored usingkcolors such that any two vertices have distinct sets of colors of them and their incident edges. In this paper, we figure out the exact value of this chromatic number of some special graphs and propose a conjecture on the upper bound of this chromatic number.

Adjacent vertex distinguishing edge-colorings and total-colorings of the Cartesian product of graphs

Let $G$ be a simple graph with vertex set $V(G)$ and edge set $E(G)$. An edge-coloring $\sigma$ of $G$ is called an adjacent vertex distinguishing edge-coloring of $G$ if $F_{\sigma}(u)\not= F_{\sigma}(v)$ for any $uv\in E(G)$, where $F_{\sigma}(u)$ denotes the set of colors of edges incident with $u$. A total-coloring $\sigma$ of $G$ is called an adjacent vertex distinguishing total-coloring of $G$ if $S_{\sigma}(u)\not= S_{\sigma}(v)$ for any $uv\in E(G)$, where $S_{\sigma}(u)$ denotes the set of colors of edges incident with $u$ together with the color assigned to $u$. The minimum number of colors required for an adjacent vertex distinguishing edge-coloring (resp. an adjacent vertex distinguishing total-coloring) of $G$ is denoted by $\chi_a^{'}(G)$ (resp. $\chi^{''}_{a}(G)$). In this paper, we provide upper bounds for these parameters of the Cartesian product $G$ &#9633 $H$ of two graphs $G$ and $H$. We also determine exact value of these parameters for the Cartesian product of a bipartite graph and a complete graph or a cycle, the Cartesian product of a complete graph and a cycle, the Cartesian product of two trees and the Cartesian product of regular graphs.

Vertex-Distinguishing Total Coloring of Ladder Graphs (<i>N</i>=0(mod 8) )

A proper total coloring of a simple graph G is called vertex distinguishing if for any two distinct vertices u and v in G, the set of colors assigned to the elements incident to u differs from the set of colors incident to v. The minimal number of colors required for a vertex distinguishing total coloring of G is called the vertex distinguishing total coloring chromatic number. In a paper, we give a “triangle compositor”, by the compositor, we proved that when n=0(mod 8) and , vertex distinguishing total chromatic number of “ladder graphs” is n.

Adjacent vertex-distinguishing edge and total chromatic numbers of hypercubes

An adjacent vertex-distinguishing edge coloring of a simple graph G is a proper edge coloring of G such that incident edge sets of any two adjacent vertices are assigned different sets of colors. A total coloring of a graph G is a coloring of both the edges and the vertices. A total coloring is proper if no two adjacent or incident elements receive the same color. An adjacent vertex-distinguishing total coloring h of a simple graph G = ( V , E ) is a proper total coloring of G such that H ( u ) ≠ H ( v ) for any two adjacent vertices u and v, where H ( u ) = { h ( w u ) | w u ∈ E ( G ) } ∪ { h ( u ) } and H ( v ) = { h ( x v ) | x v ∈ E ( G ) } ∪ { h ( v ) } . The minimum number of colors required for an adjacent vertex-distinguishing edge coloring (resp. an adjacent vertex-distinguishing total coloring) of G is called the adjacent vertex-distinguishing edge chromatic number (resp. adjacent vertex-distinguishing total chromatic number) of G and denoted by χ a v ′ ( G ) (resp. χ a t ( G ) ). In this paper, we consider the adjacent vertex-distinguishing edge chromatic number and adjacent vertex-distinguishing total chromatic number of the hypercube Q n , prove that χ a v ′ ( Q n ) = Δ ( Q n ) + 1 for n ⩾ 3 and χ a t ( Q n ) = Δ ( Q n ) + 2 for n ⩾ 2 .

On the adjacent vertex-distinguishing total chromatic numbers of the graphs with Δ (G) = 3

Let \(G=(V(G),E(G))\) be a simple graph and T(G) be the set of vertices and edges of G. Let C be a k-color set. A (proper) total k-coloring f of G is a function \(f\!: T(G)\longrightarrow C\) such that no adjacent or incident elements of T(G) receive the same color. For any \(u\in V(G)\), denote \(C(u)=\{f(u)\}\cup\{f(uv)|uv\in E(G)\}\). The total k-coloring f of G is called the adjacent vertex-distinguishing if \(C(u)\neq C(v)\) for any edge \(uv\in E(G)\). And the smallest number of colors is called the adjacent vertex-distinguishing total chromatic number \(\chi_{at}(G)\) of G. In this paper, we prove that \(\chi_{at}(G)\leq 6\) for all connected graphs with maximum degree three. This is a step towards a conjecture on the adjacent vertex-distinguishing total coloring.

D(β)-vertex-distinguishing total coloring of graphs

A new concept of the D ( β )-vertex-distinguishing total coloring of graphs, i.e., the proper total coloring such that any two vertices whose distance is not larger than β have different color sets, where the color set of a vertex is the set composed of all colors of the vertex and the edges incident to it, is proposed in this paper. The D (2)-vertex-distinguishing total colorings of some special graphs are discussed, meanwhile, a conjecture and an open problem are presented.