Graphs in which G − N[v] is a cycle for each vertex v

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Abstract

We say that G has the property P if GN[v] is a cycle for any vertex vV(G), where N[v] is the closed neighborhood of v in G. For an integer l{3,4,5,6}, let Gl be a set of graphs defined as follows:

G3={2K3}{G:both G and G are connected, and G is a triangle-free cubic graph}, where H denotes the complement of H,

G4={L(H):H is a connected triangle-free cubic graph}, where L(H) denotes the line graph of H,

G5={G20}, where G20 is the icosahedron,

G6={G(5,2)}, where G(5,2) is the Petersen graph. Furthermore, let G={G1G2Gt:Gil{3,4,5,6}Gl, t is any positive integer}. We show that a graph G has the property P if and only if GG.

Introduction

Let G=(V(G),E(G)) be a finite, simple, undirected graph. As usual, |V(G)| and |E(G)| are called the order and the size of G, respectively. If uvE(G), then we say that u is a neighbor of v in G and vice versa. For every vertex vV(G), the open neighborhood of v is the set N(v)={uV(G)|uvE(G)} and the closed neighborhood of v is the set N[v]=N(v){v}. The degree of a vertex vV(G) is d(v)=|N(v)|. The minimum degree and the maximum degree of a graph G are denoted by δ(G) and Δ(G), respectively. For any set SV(G), G[S] denotes the subgraph of G induced by S and GS denotes the graph obtained by deleting S from G. As usual, Kn,Cn,Pn denotes the complete graph, cycle, path of order n, respectively. The complement of a graph G, denoted by G, is the graph with vertex set V(G)=V(G), in which two vertices are adjacent if and only if they are not adjacent in G. The line graph of G, denoted by L(G), is the graph with vertex set V(L(G))=E(G), in which two vertices are adjacent if and only if they share a common end vertex in G. For two vertex-disjoint graphs G1 and G2, the join of G1 and G2, denoted by G1G2, is the graph obtained from G1G2 by joining each vertex of G1 to all vertices of G2. For a positive integer k, kG consists of k disjoint copies of G. We refer to [2] for undefined notations and terminology.

A graph H is said to be realizable by a graph G if every neighborhood in G induces a subgraph isomorphic to H. The problem is also referred to as the Trahtenbrot-Zykov problem, see [14], [19]. Agakisieva [1] proved that Cl is realizable by a graph if and only if 3l6. But, it was disproved by Chilton, Gould and Polimeni [11]. Laskar and Mulder [16] investigated the graphs in which every neighborhood induces a path. They characterized the graphs G with G[N(v)]Pk for k4. Graphs in which the neighborhood of each vertex is some special graphs are investigated in [6], [7], [8], [9], [12], [13], [15], [16], [17].

In this note, we investigate the graphs with somewhat opposite property to the Trahtenbrot-Zykov problem. We say that G has the property P if GN[v] is a cycle for any vertex vV(G). Since G(5,2)N[v]C6 for any vertex v, the Petersen graph G(5,2) is such a graph. In addition, since GN[v]=G[NG(v)], G has the property P if and only if G[NG(v)] is the complement of a cycle for each vertex v.

Another ingredient why we investigate the graphs with the property P comes from so called the isolation number, as the generation of domination, introduced by Caro and Hansberg [10]. It is recently studied by a number of papers [3], [4], [5], [18].

Actually, we can find more graphs with the property P by the following operation.

Lemma 1.1

Let G and H be two vertex-disjoint graphs. Then GH has property P if and only if each of G and H has property P.

The following lemma is crucial for characterizing all graphs with the property P.

Lemma 1.2

Let G be a graph with connected complement. If GN[v] is a d-regular graph for every vV(G), then G is regular, where d is a fixed nonnegative integer.

Proof

Let u and v be any two nonadjacent vertices of G. By the assumption, both GN[u] and GN[v] are d-regular. Since dG(v)=d+|N(u)N(v)|=dG(u), we have dG(v)=dG(u). It means that the degrees of any two adjacent vertices in G are equal. Therefore, G is regular, because G is connected. Thus G is regular. 

By the above lemma, if G is a graph with the property P and with the connected complement, it is regular. As we have seen before, since {v}NG(v)NG(v)=V(G), GN[v]=G[NG(v)]. The following result is immediate.

Lemma 1.3

Let G be a graph. For a vertex vV(G), GN[v]Cl if and only if G[NG[v]]Cl.

Lemma 1.4

Assume that a graph G has the property P. Then G is disconnected if and only if G2K3.

Proof

It is trivial to see that G2K3 is a disconnected graph G with the property P. Now assume that G is disconnected graph with the property P. By the definition of the property P, it follows that G has exactly two components, each of which must be K3. Thus G2K3. 

The aim of the note is to characterize all graphs with the property P.

Section snippets

The characterization

In view of Lemma 1.2, Lemma 1.4, we may assume that the graphs under consideration are connected.

Proposition 2.1

Assume that a graph G and its complement G are connected. Then GN[v]C3 for any vertex vV(G) if and only if G is a triangle-free cubic graph.

Proof

First assume that G is a triangle-free cubic graph. It is clear that for any vertex vV(G), NG(v) is an independent set of cardinality 3 in G. Thus NG(v) is a clique in G. Moreover, since NG(v)=V(G)NG[v], GNG[v]C3.

To show its necessity, assume that

Graphs and their complements have the property P

Lemma 3.1

Let G be a connected triangle-free graph with the property P. If G is connected, then GN[v] is a cycle for any vertex vV(G) if and only if GG(5,2).

Proof

It is straightforward to check that if GG(5,2), then GN[v]C6 for any vertex vV(G).

Next, assume that GN[v] is a cycle for any vertex vV(G). By Lemma 1.2, assume that G is k-regular for some integer k3. Thus GN[v]Cl, where l=n1k. Since G is triangle-free, l4.

Fix a vertex x and a neighbor y in G. Since G is triangle-free, N(x)N(y)=.

Further research

Let F denote a family of graphs. It is interesting to characterize the graphs G such that for any vertex vV(G), GN[v] is isomorphic to a member in the family F. In particular, we are interested to investigate the cases when (1) F is the family of all stars; (2) F is the family of all paths; (3) F is the family of all trees; (4) F is the family of all cliques, among others.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors are grateful to the referees for their helpful comments. The work is supported by the Key Laboratory Project of Xinjiang (2018D04017), NSFC (No. 12061073), and XJEDU2019I001.

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