Adjacent Vertex Distinguishing Edge Coloring Research Articles

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 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.

Graph coloring based channel assignment framework for rural wireless mesh networks

IEEE 802.11 based wireless mesh networks with directional antennas are expected to be a new promising technology and an economic approach for providing wireless broadband services in rural areas. In this paper, we discuss interference models and address how they can affect the design of channel assignment in rural mesh networks. We present a new channel assignment framework based on graph coloring for rural wireless mesh networks. The goal of the framework is to allow synchronously transmitting or receiving data from multiple neighbor links at the same time, and continuously doing full-duplex data transfer on every link, creating an efficient rural mesh network without interference. Channel assignment is shown to be NP-hard. We frame this channel allocation problem in terms of Adjacent Vertex Distinguishing Edge Coloring (AVDEC). Detailed assignment results on grid topology are presented and discussed. Furthermore, we design an algorithm. Finally, we evaluate the performance of the proposed algorithm through extensive simulations and show the algorithm is effective to the regular grid topologies, and the number of colors used by the algorithm is upper bounded by Δ + 1. Hence the algorithm guarantees that the number of channels available in standards such as IEEE 802.11a is sufficient to have a valid AVDEC for many grid topologies. We also evaluate the proposed algorithm for arbitrary graphs. The algorithm provides a lower upper bound on the minimum number of channels to the AVDEC index channel assignment problem.

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].

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 .