Binding Number of Corona and Join of Graphs

access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The binding number of a graph G is defined as bind(G) = min |N (S)| |S| , S ∈ F (G) where F (G) = {S ⊆ V (G) : S = ∅ and N (S) = V (G)}. This paper provides some results on the binding numbers of corona and join of graphs and characterize them in terms of independent binding set.


Introduction
In 1973, Wodall [8] introduced the concept of a binding number of a graph.He published the first known results on the binding numbers of some graphs including the binding numbers of paths, cycles, and complete graphs.While in 1985, Ronghua [7] proved Woodall's conjecture that if bind(G) ≥ 3  2 then G is pancyclic.Looking back in 1981, Kane and Mohanty [6] theorized that if bind(G) ≥ 3  2 and |V (G)| ≥ 5 then G contains a 4-cycle or a 5-cycle.While Goddard and Swart [5] published results on the binding numbers of cartesian and lexicographic product of graphs.
In this paper, we determine the binding numbers of corona and join of graphs and provide the necessary and sufficient conditions for these graphs to have the independent binding sets.
Throughout this paper, the graph we consider is simple, that is, no loop or multiple edges.We use the terminologies of Chartrand and Oellerman [3], and Harary [4].

Preliminaries
Definition 2.1 A graph G is a finite nonempty set of objects called vertices together with a (possibly empty) set of unordered pairs of distinct vertices of G called the edges.The vertex set of G is denoted by V (G), while the edge set is denoted by E(G).If G has no loops, that is, no edge joining a vertex to itself, and no multiple edges, that is, no two edges join the same pair of vertices, then the graph G is simple.
Definition 2.5 A set S of vertices in a graph G is independent if no two vertices of S are adjacent in G.A set containing independent vertices is called an independent set.An independent set S of vertices in G is called a maximal independent set if S is not a proper subset of any other independent set of vertices of G.

Definition 2.6
The corona G•H of two graphs G and H is the graph obtained by taking one copy of G of order n and n copies H i of H, and then joining the ith vertex of G to every vertex of H i .

Definition 2.7
The join G + H of two graphs G and H is the graph with Definition 2.8 [8] Let G be a graph with vertex set V (G) and define The binding set of G is any set Proposition 2.9 [8] For any graph G on n vertices and with minimum degree The binding set of

Binding Number of Corona of Graphs
Theorem 3.1 Given H a connected or trivial graph and K a connected graph with Proof : Let's prove when H and K are connected and the case when which violates Proposition 2.9.Thus, A does not give the binding number.So now, let's consider the set Theorem 3.2 Let H be a disconnected graph and K a connected graph.Then bind(H ) is largest and M t is the smallest among the disconnected subgraphs of the ith copy of K.
Proof : Let M 1 , M 2 , . . ., M s be the disconnected subgraphs of K and M t be the smallest subgraph.Define Proof : Let G be a corona of graphs.Suppose that G has independent binding set.Let S be a binding set of G. Then S is independent.It follows that the elements of S are the nonadjacent vertices of G. Thus, G has a subgraph tree.
Conversely, suppose that |S| .Thus, X is the independent binding set.Therefore, G has independent binding set.

Binding Number of Join of Graphs
where S * and T * are the maximum binding sets of G and H respectively.
Proof : Let G and H be graphs and let S, T ∈ F (G + H).We have to show that S ∈ F (G) or S ∈ F (H) and similar argument will be applied to T .Claim.
Thus, if S ∈ F (G + H) then by the claim, S ∈ F (G) or S ∈ F (H).We just assume that S ∈ F (G). Then |S| can be obtained for some choice of S, say S * .Hence, With the same argument above we have, where S is a maximum binding set of G.

|S|
. This implies that S is also a binding set of G + H.In either case G + H has independent binding set.

Theorem 3 . 3
Let G = H • K with H a connected graph of order m and K a disconnected graph of order n with no trivial subgraph.Then bind

Remark 3 . 4 1 . 3 . 5 Theorem 3 . 6
is not necessarily of maximum order.Then by the same argument in the proof of Result 3.1, |T | ≤ |S| and |N (S)| |S| ≥ |N (T )| |T | .Thus, T gives the binding number of G and therefore, bind(G) = n(m+1)+|N(Y )| n(m+1)+|Y |−1 .Let G = H • K with H a connected graph of order m and K a disconnected graph of order n.If K has a trivial subgraph then bind(G) = Theorem Let H and K be graphs of order m and n respectively.If G = H • K has independent binding set S then bind(G) = 1 |S * | where S * is the largest subset of S in the ith copy of K.Proof : Suppose that G = H • K has independent binding set.Let S be a binding set of G. Then S consists of nonadjacent vertices in G. Hence, S ⊆ V (K v ) where v ∈ V (H).It follows that N(S) ⊆ V (H).Let S * be a subset of S in the ith copy of K of maximum order.Since there are m timesK v , we have |S| = m|S * | where 1 ≤ |S * | ≤ n and |N (S)| = m.Therefore, bind(G) = |N (S)| |S| = m m|S * | = 1 |S * | .Let G be a corona of graphs.Then G has independent binding set if and only if G has a subgraph tree.

Theorem 4 . 1
Let G and H be graphs.Then bind

ProofCorollary 4 . 3
: Let G be a graph and S a binding set of G + K n .If G is trivial then we can take S = G and the conclusion follows.If G is complete then either S ∈ F (G) or S ∈ F (K n ).Suppose that G is nontrivial and noncomplete.Then S ∈ F (G) and by Result 4.1, bind(G+K n ) = |N G (S)|+|V (Kn)| |S| = |N (S)| |S| + |V (Kn)| |S| = bind(G) + n |S| where S is a maximum binding set of G.For n ≥ 3, bind (S)|+|V (H)| |S| which shows that S is a binding set of G + H.If H has independent binding set S then we have bindG + H) = |N G (S)|+|V (G)| Proof : Let F n be a fan with n ≥ 3. SinceF n = P n + K 1 , by Result 4.2, bind(F n ) = bind(P n + K 1 ) = bind(P n ) + 1|S| where S is a binding set of P n .By Proposition 2.11 (c), bind( n is odd. 5.5If G is a graph with independent binding set then G + K n has independent binding set.Proof : If G is trivial or complete graph then the conclusion follows.Suppose that G is nontrivial and noncomplete graph.Let S be a binding set of G.We have to show that S is a binding set of G + K n .Suppose that S * is a binding set of G+K n .Using parallel argument in the proof of Theorem 4.1, S By Result 4.6, S is a binding set of either G or H.But S is independent, it follows that G or H has independent binding set.Conversely, suppose that either G or H has independent binding set.If G has independent binding set then by Result 4.2, bind(G + H) = |N H * ∈ F (G) or S * ∈ F (K n ).If S * ∈ F (K n ) then S * must be a singleton by Proposition 2.11.Since S * need not be a singleton, S * ∈ F (G). Consequently, S = S * .Therefore, G + K n has independent binding set.Theorem 4.6 Let G and H be graphs.If S is a binding set of G + H then S is a binding set of either G or H. Proof : Let G and H be graphs.Suppose that S is a binding set of G + H. Then by Result 4.2, S ∈ F (G) or S ∈ F (H).Note that the minimum of |N (T )| |T | for any T ∈ F (G) or T ∈ F (H) remains if we compute bind(G \ H) = bind(G) or bind(H \ G) = bind(H).Taking S = T , then S is a binding set of either G or H. Theorem 4.7 Let G and H be graphs.Then G + H has independent binding set if and only if either G or H has independent binding set.Proof : Let G and H be graphs.Suppose that G + H has independent binding set.Let S be a binding set of G + H.