Adjacent Vertex Distinguishing Edge Research Articles

Adjacent vertex distinguishing edge choosability of 1-planar graphs with maximum degree at least 23

For a simple graph G, an adjacent vertex distinguishing edge k-coloring of G is a mapping ϕ: E(G)→{1,2,…,k} such that ϕ(e1)≠ϕ(e2) for every adjacent edges e1 and e2 in E(G) and Cϕ(u)≠Cϕ(v) for each edge uv∈E(G), where Cϕ(v) (or Cϕ(u)) denotes the set of colors assigned to the edges incident with v (or u). For each edge e∈E(G), let L(e) be a list of possible colors that can be used on e. If, whenever we give a list assignment L={L(e)||L(e)|=k,e∈E(G)}, there exists an adjacent vertex distinguishing edge k-coloring ϕ such that ϕ(e)∈L(e) for each edge e∈E(G), then we say that ϕ is a list adjacent vertex distinguishing edge k-coloring. The smallest k for which such a coloring exists is called the adjacent vertex distinguishing edge choosability of G, denoted by cha′(G). A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. There is almost no result yet about cha′(G) if G is a 1-planar graph. We prove that cha′(G)≤Δ+3 for every 1-planar graph G with maximum degree Δ≥23.

The adjacent vertex distinguishing edge choosability of planar graphs with maximum degree at least 11

An adjacent vertex distinguishing edge coloring is a proper edge coloring such that any two adjacent vertices have distinct sets consisting of colors of their incident edges. It was conjectured that if G is a connected graph with at least 6 vertices, then χa′(G)≤Δ(G)+2. In this paper, we consider the list version of adjacent vertex distinguishing edge coloring and show that if G is a planar graph without isolated edges, then cha′(G)≤max{13,Δ(G)+2}, where cha′(G) is the adjacent vertex distinguishing edge choosability. This result implies that the above conjecture holds for planar graphs with maximum degree at least 11. Our approach is based on the Combinatorial Nullstellensatz and the discharging method.

Adjacent vertex distinguishing edge coloring of the semistrong product of paths

An adjacent vertex distinguishing edge coloring of a 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 required for an adjacent vertex distinguishing edge coloring of G is called the adjacent vertex distinguishing edge chromatic number of G, and denoted by . In this paper, use the method of structural chromatic and the method of proof to the contrary and obtain the adjacent vertex distinguishing edge chromatic numbers of the semi-strong product of two paths.

A Novel Algorithm for Adjacent Vertex-Distinguishing Edge Coloring of Large-scale Random Graphs

A Novel Algorithm for Adjacent Vertex-Distinguishing Edge Coloring of Large-scale Random Graphs

Adjacent vertex distinguishing edge coloring of planar graphs without 3-cycles

The adjacent vertex distinguishing edge coloring of a graph [Formula: see text] is a proper edge coloring of [Formula: see text] such that the color sets of any pair of adjacent vertices are distinct. The minimum number of colors required for an adjacent vertex distinguishing edge coloring of [Formula: see text] is denoted by [Formula: see text]. It is observed that [Formula: see text] when [Formula: see text] contains two adjacent vertices of degree [Formula: see text]. In this paper, we prove that if [Formula: see text] is a planar graph without 3-cycles, then [Formula: see text]. Furthermore, we characterize the adjacent vertex distinguishing chromatic index for planar graphs of [Formula: see text] and without 3-cycles. This improves a result from [D. Huang, Z. Miao and W. Wang, Adjacent vertex distinguishing indices of planar graphs without 3-cycles, Discrete Math. 338 (2015) 139–148] that established [Formula: see text] for planar graphs without 3-cycles.

Progress on the Adjacent Vertex Distinguishing Edge Coloring Conjecture

A proper edge colouring of a graph is adjacent vertex distinguishing if no two adjacent vertices see the same set of colours. Using a clever application of the Local Lemma, Hatami (2005) proved that every graph with maximum degree $\Delta$ and no isolated edge has an adjacent vertex distinguishing edge colouring with $\Delta + 300$ colours, provided $\Delta$ is large enough. We show that this bound can be reduced to $\Delta + 19$. This is motivated by the conjecture of Zhang, Liu, and Wang (2002) that $\Delta + 2$ colours are enough for $\Delta \geq 3$.

Adjacent Vertex Distinguishing Edge Coloring of Planar Graphs Without 4-Cycles

The adjacent vertex distinguishing edge coloring of a graph G is a proper edge coloring of G such that the edge coloring set on any pair of adjacent vertices is distinct. The minimum number of colors required for an adjacent vertex distinguishing edge coloring of G is denoted by $$\chi _{a}'(G)$$. It is observed that $$\chi _a'(G)\ge \Delta (G)+1$$ when G contains two adjacent vertices of degree $$\Delta (G)$$. In this paper, we prove that if G is a planar graph without 4-cycles, then $$\chi _a'(G)\le \max \{9,\Delta (G)+1\}$$.

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.

Asymptotically optimal bound on the adjacent vertex distinguishing edge choice number

AbstractAn adjacent vertex distinguishing edge coloring of a graph G without isolated edges is its proper edge coloring such that no pair of adjacent vertices meets the same set of colors in G. We show that such coloring can be chosen from any set of lists associated to the edges of G as long as the size of every list is at least , where Δ is the maximum degree of G and C is a constant. The proof is probabilistic. The same is true in the environment of total colorings.

Upper bounds for adjacent vertex-distinguishing edge coloring

An adjacent vertex-distinguishing edge coloring of a graph is a proper edge coloring such that no pair of adjacent vertices meets the same set of colors. The adjacent vertex-distinguishing edge chromatic number is the minimum number of colors required for an adjacent vertex-distinguishing edge coloring, denoted as $$\chi '_{as}(G)$$ . In this paper, we prove that for a connected graph G with maximum degree $$\Delta \ge 3$$ , $$\chi '_{as}(G)\le 3\Delta -1$$ , which proves the previous upper bound. We also prove that for a graph G with maximum degree $$\Delta \ge 458$$ and minimum degree $$\delta \ge 8\sqrt{\Delta ln \Delta }$$ , $$\chi '_{as}(G)\le \Delta +1+5\sqrt{\Delta ln \Delta }$$ .

Adjacent vertex distinguishing index of bipartite planar graphs

A proper edge coloring of a graph $G$ is called adjacent vertex distinguishing, if any pair of adjacent vertices meet distinct color sets. The adjacent vertex distinguishing index of $G$, denoted by $\chi_a(G)$, is the smallest integer $k$ such that $G$ has an adjacent vertex distinguishing edge $k$-coloring. In this paper, we show that every bipartite planar graph $G$ with maximum degree $\Delta\ge 7$ and without isolated edges has $\chi_{a

On the adjacent vertex-distinguishing equitable edge coloring of graphs

Let G(V,E) be a graph. A k-adjacent vertex-distinguishing equatable edge coloring of G, k-AVEEC for short, is a proper edge coloring f if (1) C(u) ≠ C(v) for ∨uv ∈ E(G), where C(u) = {f(uv)|uv ∈ E}, and (2) for any i, j = 1, 2, … k, we have ‖Ei| − |Ej‖ ≤ 1, where Ei = {e|e ∈ E(G) and f(e) = i}. χ′ave(G) =min{k| there exists a k-AVEEC of G} is called the adjacent vertex-distinguishing equitable edge chromatic number ofG. In this paper, we obtain the χ′ave(G) of some special graphs and present a conjecture.

Adjacent vertex distinguishing edge colorings of planar graphs with girth at least five

An adjacent vertex distinguishing edge coloring of a graph $$G$$ is a proper edge coloring of $$G$$ such that any pair of adjacent vertices admit different sets of colors. The minimum number of colors required for such a coloring of $$G$$ is denoted by $$\chi ^{\prime }_{a}(G)$$ . In this paper, we prove that if $$G$$ is a planar graph with girth at least 5 and $$G$$ is not a 5-cycle, then $$\chi ^{\prime }_{a}(G)\le \Delta +2$$ , where $$\Delta $$ is the maximum degree of $$G$$ . This confirms partially a conjecture in Zhang et al. (Appl Math Lett 15:623---626, 2002).

Adjacent vertex-distinguishing edge coloring of graphs with maximum degree Δ

An adjacent vertex-distinguishing edge coloring, or avd-coloring, of a graph G is a proper edge coloring of G such that no pair of adjacent vertices meets the same set of colors. Let $\operatorname {mad}(G)$ and Δ(G) denote the maximum average degree and the maximum degree of a graph G, respectively. In this paper, we prove that every graph G with Δ(G)?5 and $\operatorname{mad}(G) < 3-\frac {2}{\Delta}$ can be avd-colored with Δ(G)+1 colors. This completes a result of Wang and Wang (J. Comb. Optim. 19:471---485, 2010).

Adjacent vertex-distinguishing edge coloring of graphs with maximum degree at least five

Abstract An adjacent vertex-distinguishing edge coloring, or avd-coloring, of a graph G is a proper edge coloring of G such that no pair of adjacent vertices meets the same set of colors. Let mad ( G ) and Δ ( G ) denote the maximum average degree and the maximum degree of a graph G, respectively. In this note, we prove that every graph G with Δ ( G ) ⩾ 5 and mad ( G ) 13 5 can be avd-colored with Δ ( G ) + 1 colors. This strengthens a result of Wang and Wang [W. Wang and Y. Wang, Adjacent vertex distinguishing edge-colorings of graphs with smaller maximum average degree, J. Comb. Optim., 19:471–485, 2010].

On the adjacent vertex distinguishing edge colourings of graphs

A k-adjacent vertex distinguishing edge colouring or a k-avd-colouring of a graph G is a proper k-edge colouring of G such that no pair of adjacent vertices meets the same set of colours. The avd-chromatic number, denoted by χ′a(G), is the minimum number of colours needed in an avd-colouring of G. It is proved that for any connected 3-colourable Hamiltonian graph G, we have χ′a(G)≤Δ+3.

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 .