eu FEEDBACK NUMBERS OF FLOWER SNARK AND RELATED GRAPHS

Let G = (V,E) be a graph or digraph without multiple edges, with vertex set V (G) and edge set E(G). A subset F ⊂ V (G) is called a feedback vertex set if the subgraph G − F is acyclic, that is, if G − F is a forest. The minimum cardinality of a feedback vertex set is called the feedback number (or decycling number proposed first by Beineke and Vandell [1]) of G. A feedback vertex set of this cardinality is called a minimum feedback vertex set.


Introduction
Let G = (V, E) be a graph or digraph without multiple edges, with vertex set V (G) and edge set E(G).A subset F ⊂ V (G) is called a feedback vertex set if the subgraph G − F is acyclic, that is, if G − F is a forest.The minimum cardinality of a feedback vertex set is called the feedback number (or decycling number proposed first by Beineke and Vandell [1]) of G.A feedback vertex set of this cardinality is called a minimum feedback vertex set.
Apart from its graph-theoretical interest, the minimum feedback vertex set problem has important application to several fields.For example, the problems are in operating systems to resource allocation mechanisms that prevent deadlocks [2], in artificial intelligence to the constraint satisfaction problem and Bayesian inference, in synchronous distributed systems to the study of monopolies and in optical networks to converters placement problem(see [3] and [4]).
Determining the feedback number is quite difficult even for some elementary graphs.However, the problem has been studied for some special graphs and digraphs, such as hypercubes, meshes, toroids, butterflies, cube-connected cycles, directed split-stars (see [3]- [13]).In fact, the minimum feedback set problem is known to be N P -hard for general graphs [14] and the best known approximation algorithm is one with an approximation ratio two [5].
Many research of literatures have been studied about flower snark and its related graphs.For example, Zheng.[15] has been studied the crossing number of flower snark and its related graph; Xi. [16] has been studied super vertexmagic total labelings of flower snark and related graphs; Mo.hammad [17] and Tong.[18] have been studied labeling of flower snark and related graphs; In addition, the adjacent vertex distinguishing incidence coloring number of flower graphs has been studied in area of mathematics.But, there is little research done so far about the feedback number of flower snark and related graphs.
In this paper, we consider the feedback number of flower snark and related graphs H n .Let f (H n ) denote the feedback number of H n , we proves that: where the vertex labels are read modulo n.
Let H n be a graph obtained from G n by replaceing the edges b n−1 b 0 and c n−1 c 0 with b n−1 c 0 and c n−1 b 0 respectively.For odd n ≥ 5, H n is called a Snark, namely Flower Snark [19] and [20].G n and H n (n = 3 or even n ≥ 4) are called the related graphs of Flower Snark [21].
In this paper, we denote H n as Flower Snark and its related graphs.By the definition of H n , the set of V (H n ) can be divided into three parts as follows: Obviously, H 1 is the only one cycle which contains all the vertices for {a i }, H 2 is the only one cycle which contains all the vertices for {b j } and {c k }.
Proof.Since H 1 is a cycle, then we only delete any one vertex, say a n−1 , to obtain an acyclic subgraph of Similarly, in H 2 we only delete any one vertex, say c n−1 , to obtain an acyclic subgraph denoted by Since By the definition of feedback vertex set, F n is a feedback vertex set of H n .

Feedback Number of H n
Lemma 2. For feedback vertex set F n in a graph G = (V, E) with maximum degree ∆, it holds that The feedback vertex set in H n is of size at least: Proof.Noting that H n (n ≥ 3) has 4n vertices and 6n edges,the maximum degree is 3, by Lemma 2, we immediately obtain the lower bound Theorem 1. Feedback vertex number of flower snark and related graphs Proof.For n ≥ 3, by definition, it is easy to find that Since F n is a feedback vertex set, then we have The theorem holds.
In Figure 2, we show the feedback vertex sets of several H n graphs with small n, where the vertices of feedback vertex sets are in red, the acyclic graph are in blue.

8 Figure 2 :
Figure 2: Flower snark and its related graph of H n