( , ) = { ( , ), ( , ), ( , ), ( , )}. If ( , ) = then these edges are called neighbor edges. In this paper, we investigate the ′( , )- **edge** **coloring** of **some** **graphs**. The ′( , )-**edge** **coloring** of a graph is an assignment of non-negative integers to the edges and of such that | ( ) − ( )| ≥ if ( , ) = and | ( ) − ( )| ≥ if ( , ) = . No restriction is placed on colors assigned to edges at distance **2** or more. We also define the ′( , )- **edge** **coloring** number, ′ ( )of **some** **graphs** viz. cycles, complete **graphs**, wheel **graphs**, complete bipartite **graphs**, fan **graphs** and

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Abstract: The **L**(**2**, **1**)-labeling (or distance two labeling) of a graph G is an integer labeling of G in which two vertices at distance one from each other must have labels differing by at least **2** and those vertices at distance two must differ by at least **1**. The **L**(**2**, **1**)-labeling number of G is the smallest number k such that G has an **L**(**2**, **1**)-labeling with maximum of f(v) is equal to k, where v belongs to vertex set of G. In this paper, upper bound for the **L**(**2**, **1**)-labeling number for the β-product of two **graphs** has been obtained in terms of the maximum degrees of the **graphs** involved.

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Graph products play an important role in connecting many useful networks. Klavzar and Spacepan [9] have shown that ∆ **2** -conjecture holds for **graphs** that are direct or strong products of nontrivial **graphs**. After that Shao, et al. [13] have improved bounds on the **L**(**2**, **1**)-labeling number of direct and strong product of nontrivial **graphs** with refined approaches. Shao and Shang [15] also consider the graph formed by the Cartesian sum of **graphs** and prove that the λ -number of **L**(**2**, **1**)-labeling of this graph satisfies the

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The line graph of G is denoted by **L** ( G ) and the basic properties of line **graphs** are found in **some** textbooks, for example, in [18]. The iterated line **graphs** of G are then defined recur- sively as **L** **2** ( G ) = **L** ( **L** ( G )) , **L** 3 ( G ) = **L** ( **L** **2** ( G )) , ..., **L** k ( G ) = **L** ( **L** k − **1** ( G )) . The basic properties of iterated line graph sequences are summarized in the articles [6, 7]. Authors in [21] show that if G is a regular graph of order n and of degree r ≥ 3, then for each k ≥ **2**, E ( **L** k ( G )) depends solely on n and r. In particular, E ( **L** **2** ( G )) = 2nr ( r − **2** ) . In [14] authors establish relations between the energy of the line graph of a graph G and the energies associated with the Laplacian and signless Laplacian matrices of G.

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ψ . The well-known AVDTC conjecture, made by Zhang et al. [8] says that every simple graph G has χ avt ( G ) ≤ ∆ ( G ) + 3 . AVDTC of tensor product of **graphs** are discussed in litrature [6,7]. (2,1)-total labelling of cactus **graphs** and adjacent vertex distinguishing **edge** colouring of cactus **graphs** are discussed in literature [3,4]. Throughout the paper, we denote the path graph, cycle graph, complete graph, star graph, sun let, fan graph, double star and friendship graph by P n , C n , K n , K **1**, n , S **2** n

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One can easily see that each simple graph can be properly col- ored with **2** ∆ − **1** colors in time O ( ∆ m ) , where ∆ is the maximum node degree in the graph. An astonishing theorem by Vizing states that any simple graph can be colored either by ∆ (class **1**) or ∆ + **1** (class **2**) colors. **1** Holyer [14] proved that **edge** **coloring** is N P - hard on general **graphs** and hence all known exact algorithmic endeavours require exponential time. However, Misra & Gries [19] provided a constructive proof of Vizing's theorem. The resulting algorithm inds an **coloring** with at most ∆ + **1** colors in time O ( nm ) and makes use of a sophisticated procedure called a fan rotation. Oftentimes, restrictions to speciic graph classes lead to more ei- cient algorithms since one can leverage structural properties. For ∆ -**edge**-colorable bipartite **graphs**, algorithms with running time O(m log m) by Alon [**1**], O( ∆ m) by Schrijver and O(m log ∆ ) by Cole, Ost and Schirra [8] have been proposed.

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In Sections 3.2.1 and 3.2.2 we briefly covered **some** methods we used to generate cycle-free colorings using Constraint Satisfaction Problems. One question that remained unanswered with respect to generating colorings using SAT solvers is for which cases is this method faster than direct enumeration. The answer might lie in the clause density of the formulas that are generated by this approach, since the SAT solving community is often concerned with this parameter [DW06]. Conversely, proper benchmarking of several SAT solvers could reveal what implementation—if any—is better suited for this problem.

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Proof. It is an easy matter to see that G H = (G × H) (G ∧ H ). And so, χ (G H) ≤ χ (G × H)+ χ (G ∧ H). Since at least one of G and H is of class **1**, by Theorems 1.1 and 1.7, χ (G × H ) ≤ ∆(G) + ∆(H ) and χ (G ∧ H ) ≤ ∆(G)∆(H ). Therefore, χ (G H ) ≤ ∆(G) + ∆(H) + ∆(G)∆(H) = ∆(G H ). The proof is complete. The following result is a straightforward consequence of Theorem 2.1 and the fact that a regular graph is of class **1** if and only if it is **1**-factorable.

Let e be any given **edge** of graph G = ( V E , ) . The number of sets which consist of e and two other edges ad- jacent to e at the same vertex, is less than ∆ **2** . For every k ≥ 3 , no **edge** lies in more than ∆ k− **2** cycles of length k. For every node E X ∈ V H ( ) , let x be the number of edges contained in X. Lemma 3 tells us that the number of nodes of type I, II, III and IV adjacent to E X in graph H is no more than x∆ **2** , x ∆ k − **2** , x ∆ **l** − **2** and

Let G=(V,E) be a graph and let g be a function that assigns to each **edge** a set of colors chosen from the power set of {1,2} i.e., g:E(G)→ {1,2}. If for each **edge** E(G) such that g( we have ⋃ = {1,2},then g is called **2**-Rainbow **edge** domination function(2REDF) and the weight w(g) of a function is defined as w(g) = ∑ .

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If we change “3-representable” by “word-representable” in Theo- rem 12 we would obtain a weaker, but clearly still true statement, which is not hard to prove directly via semi-transitive orientations. In- deed, each path of length at least 3 added instead of an **edge** e can be oriented in a “blocking” way, so that there would be no directed path between e’s endpoints. Thus, **edge** subdivision does not preserve the property of being non-word-representable. The following theorem shows that **edge** subdivision may be preserved on **some** subclasses of word-representable **graphs**, but not on the others.

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Definition **1** [5] An **edge** even graceful labeling of a graph G ( V ( G ) , E ( G )) with p = |V(G)| vertices and q = |E(G)| edges is a bijective mapping f of the **edge** set E(G) into the set {**2**, 4, 6, · · · , 2q } such that the induced mapping f ∗ : V (G) → {0, **2**, 4, · · · , 2q }, given by: f ∗ (x) = xy∈E(G) f (xy)

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3 √ 30 11 ) Consider the T(p,q) nanotube. The number of edges of graph T(p,q), line graph **L**(T (p, q)) and total graph T (T (p, q)) are 6pq − p,12pq − 4p and 30pq − 7p, respectively. If we consider to the edges of T(p,q) in T (T (p, q)), there exist 4p edges with endpoints which have degrees 4 and 6, and 6pq − 5p edges with endpoints which have degree 6.

Abstract. Let G = (V, E) be a simple connected and undirected graph. A set F of edges in G is called an **edge** dominating set if every **edge** e in E − F is adjacent to at least one **edge** in F . The **edge** domination number γ 0 (G) of G is the minimum cardinality of an **edge** dominating set of G. The shadow graph of G, denoted D **2** (G) is the graph

The chromatic polynomial counts the number of ways a graph can be colored using no more than a given number of colors. For example, using three colors, the graph in the image to the right can be colored in 12 ways. With only two colors, it cannot be colored at all. With four colors, it can be colored in 24 + 4⋅12 = 72 ways: using all four colors, there are 4! = 24 valid colorings (every assignment of four colors to any 4-vertex graph is a proper **coloring**); and for every choice of three of the four colors, there are 12 valid 3-colorings. So, for the graph in the example, a table of the number of valid colorings would start like this:

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The corona [9] of two **graphs** G and H is the graph obtained by taking one copy of G, |V(G)| copies of H and joining i-th vertex of G to every vertex in the i-th copy of H. The **edge** corona [17] of two **graphs** G and H denoted by G◊H is obtained by taking one copy of G and |E(G)| copies of H and joining end vertices of i-th **edge** of G to every vertex in the i-th copy of H. Various researchers studied the topological indices of the corona of two **graphs**, for example see [23, 27]. Motivated by these works, in this paper we compute **some** topological indices of **edge** corona of two **graphs**. In Section **2**, we derive formulas for the Wiener index and Zagreb indices of **edge** corona of two **graphs**. As special cases of these formulas we express Wiener index and Zagreb indices of **edge** corona P n ◊H, C n ◊H and

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d v of a vertex v is the number of edges in E incident to v. We denote the maximum degree of G by ∆ ( ) G or simply by ∆ . A cactus G can be represented by an under tree T, which is a rooted tree. In the underlay tree T of G, each node represents a block which is either a bridge (**edge**) of G or an elementary cycle of G. If there is an **edge** between nodes b **1** and b **2** of T, then bridges or cycles of G represented by b **1** and b **2** share exactly one vertex in G. Each node b of T corresponds to a subgraph G b of G induced by all bridges and cycles represented by the nodes that are descendants of b in T. Figure **2**(a) depicts the subgraph

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to arbitrary connected **graphs** of order n by Gary Chartrand, Ladislav Nebesky, and Ping Zhang .A Hamiltonian **coloring** of a connected graph G of order n is a vertex **coloring** c such that, D (u,v) + |c(v)| > n – **1**, for every tow distinct vertices u and v of G. the largest color assigned to a vertex of G by c is called the value of c and is denoted by hc(c). The Hamiltonian chromatic number hc (G) is the smallest value among all Hamiltonian colorings of G.

end vertices of 𝐺 are assigned by one of the 𝑟 colors. So there can be at most 𝑟 **2** edges whose absolute diﬀerences of the any two distinct edges are distinct. This is a contradiction to the fact that 𝐺 has more than 𝑘 **2** edges. Hence, 𝜒 𝑆𝑇 𝐺 > 𝑘.

and means f (e) = 0 and since f : E(G) → {0,1,2} is –function, then any **edge** must be adjacent to **edge** h , i.e.,f (h) = **2** and in the function g : E(G) → {1,2}, if then g( h ) = {1,2}. Hence g : E(G) → {1,2} is a **2**-rainbow **edge** domination function in G with the weight W(g) that means

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