The Number of Blocks of a Graph with Given Minimum Degree

<jats:p>A block of a graph is a nonseparable maximal subgraph of the graph. We denote by <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1">
                        <mi>b</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>G</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> the number of block of a graph <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2">
                        <mi>G</mi>
                     </math>
                  </jats:inline-formula>. We show that, for a connected graph <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3">
                        <mi>G</mi>
                     </math>
                  </jats:inline-formula> of order <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4">
                        <mi>n</mi>
                     </math>
                  </jats:inline-formula> with minimum degree <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5">
                        <mi>k</mi>
                        <mo>≥</mo>
                        <mn>1</mn>
                     </math>
                  </jats:inline-formula>, <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6">
                        <mi>b</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>G</mi>
                           </mrow>
                        </mfenced>
                        <mo><</mo>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mfenced open="(" close=")" separators="|">
                                 <mrow>
                                    <mn>2</mn>
                                    <mi>k</mi>
                                    <mo>−</mo>
                                    <mn>3</mn>
                                 </mrow>
                              </mfenced>
                              <mo>/</mo>
                              <mrow>
                                 <mfenced open="(" close=")" separators="|">
                                    <mrow>
                                       <msup>
                                          <mrow>
                                             <mi>k</mi>
                                          </mrow>
                                          <mrow>
                                             <mn>2</mn>
                                          </mrow>
                                       </msup>
                                       <mo>−</mo>
                                       <mi>k</mi>
                                       <mo>−</mo>
                                       <mn>1</mn>
                                    </mrow>
                                 </mfenced>
                              </mrow>
                           </mrow>
                        </mfenced>
                        <mi>n</mi>
                     </math>
                  </jats:inline-formula>. The bound is asymptotically tight. In addition, for a connected cubic graph <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M7">
                        <mi>G</mi>
                     </math>
                  </jats:inline-formula> of order <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M8">
                        <mi>n</mi>
                        <mo>≥</mo>
                        <mn>14</mn>
                     </math>
                  </jats:inline-formula>, <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M9">
                        <mi>b</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>G</mi>
                           </mrow>
                        </mfenced>
                        <mo>≤</mo>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>n</mi>
                              <mo>/</mo>
                              <mn>2</mn>
                           </mrow>
                        </mfenced>
                        <mo>−</mo>
                        <mn>2</mn>
                     </math>
                  </jats:inline-formula>. The bound is tight.</jats:p>


Introduction
We consider finite, undirected, simple graphs only. Let G � (V(G), E(G)) be a graph. e numbers of vertices and edges of G are called the order and the size of G and denoted by v(G) and e(G), respectively. A vertex v is called a cut vertex if com(G − v) > com(G), where com(G) denotes the number of components of G. c(G) denotes the number of cut vertices of G. Rao [1] proved that, for a connected graph G of order n and size m, characterized all extremal graphs. Rao and Rao [2] solved the corresponding problem for a strong digraph. Later, Achuthan and Rao [3] determined the maximum number of cut edges in a connected d-regular graph of order p. Let f(n, d) � max c(G): G { is a connected k-regular graph of order n}. Rao [4] determined f(n, d) for d ≤ 4. Nirmala and Rao [5] showed that f(n, d) � ((2n − d − 5)/(d + 1)) − 1 or ((2n − d − 5)/(d + 1)) − 2 for odd d ≥ 5 and have obtained an upper bound for f(n, d) for even d ≥ 6.
Alberten and Berman [6] proved that, for a graph G of order n and minimum degree k ≥ 2, is bound is asymptotically tight. Hopkins and Staton [7] showed that every connected graph of order n contains no more than (r/(2r − 2))n cut vertices of degree r. Some related results are referred to [8,9].
A separation of a connected graph is a decomposition of the graph into two nonempty connected subgraphs which have just one vertex in common. e common vertex is called a separating vertex of the graph. Since the graph G under consideration is simple, v ∈ V(G) is a separating vertex if and only if it is a cut vertex. A block of a graph is a nonseparable maximal subgraph of the graph. We denote by b(G) the number of blocks of a graph G.
It is clear that any two blocks of a graph have at most one vertex in common. Recall that the block tree B(G) of G is the bipartite graph with bipartition (B, S), where B is the set of blocks of G and S, the set of separating vertices of G, and a block B, and a separating vertex v is joined by an edge in B(G) if and only if B contains v. It is easy to see that if G is connected, B(G) is a tree. Each leaf of B(G) corresponds to an end block of G.
Inspired from the bound for the cut vertices, in the present paper, we consider the upper bound for the number of blocks, a connected graph of order n with given minimum degree. Let us begin with two easy cases when δ(G) � 1 and δ(G) � 2.
Proof. Our proof is induction on n. If n � 2, then G � K 2 ; thus, the result holds. Next, we assume that n ≥ 3. If G has no cut vertex, then b(G) � 1 < n − 1. Now suppose G has a cut vertex. Let B be an end block of G and v be the cut vertex, which belongs to B.
By the induction hypothesis, G ′ is a tree, implying that G is a tree.
On the contrary, if G is a tree, clearly, b(G) � n − 1. □ Proposition 2. For a connected graph G of order n ≥ 4 with δ(G) ≥ 2, b(G) ≤ n − 3, with equality if and only if G is the graph obtained from P n− 4 identifying each end with a vertex of separate K 3 , as given in Figure 1.
Proof. If G has no cut vertex, the result holds trivially. Next, we assume that G has cut vertices, and thus, it has at least two end blocks and n ≥ 5. Let B 1 , . . . , B t be all end blocks of G. Let c i be the cut vertex of G, which belongs to B i for each Otherwise, G has at least two cut vertices. It follows that (3) From the above, b(G) � n − 3 if and only if t � 2 and G ′ � P n− 4 , B 1 � K 3 � B 2 , as we promised.

□
It is clear that, for a graph G of order n, b(G) decreases when δ(G) increases. For a connected graph G of order n and minimum degree at least k, we have the following result, which is asymptotically best possible.
We show that the bound in the above theorem is asymptotically best possible. Let k ≥ 3. Consider a tree T of order p with each vertex having degree k or 1. By the handshaking lemma, the number of leaves of this tree is Let G be the graph obtained from identifying each leaf of T with a vertex of a clique of order k + 1 separately. erefore, However, as p gets larger, b(G)/v(G) gets arbitrarily close to What happens for the k-regular graphs? e situation becomes complicated. We are just able to get an exact bound for a cubic graph G of order n: b(G) < n/2 ( eorem 2), whereas by eorem 1, we have b(G) < (3/5)n for a connected graph G of order n with δ(G) ≥ 3.

Theorem 2. For a connected cubic graph
e bound is sharp.
To see the sharpness of the bound, we denote by K * 4 the graph obtained from K 4 by replacing an edge with a path of length two, as drawn in Figure 2.
e graphs G n achieve the upper bound in eorem 2, which are classified into three types in terms of n ≡ 4 (mod 6), n ≡ 0 (mod 6), and n ≡ 2 (mod 6), respectively.
For an integer n ≡ 4 (mod 6), k � (n − 4)/3 is an even integer. Let T k be a tree in which every vertex has degree 1 or 3. It is clear that T k has exactly (k + 1)/2 vertices of degree 1 (leaves) and (k − 1)/2 vertices of degree 3. Let G n be a graph obtained from identifying each leaf of T k with the vertex of degree two of a separate K * 4 , as shown in Figure 3. For an integer n ≡ 0 (mod 6), let G n be a cubic graph obtained from a graph G n− 2 by replacing a vertex of degree three (not belongs to any K * 4 ) with a triangle, as shown in Figure 3.
For an integer n ≡ 2 (mod 6), let G n be a cubic graph obtained from a graph G n− 4 by inserting a K 4 − e into an edge of G n− 4 (not belongs to any K * 4 ), as shown in Figure 3. It can be checked that v(G n ) � n and b(G n ) � (n/2) − 2 for any graph G n constructed as above.
Proof of Claim 1. If it is not, let B be an end block of G. Let G ′ be the graph obtained from G by replacing B with B ′ of order k + 1.
Combining (11) with the fact that b(G) � b(G ′ ) + 1, we have a contradiction.  Discrete Dynamics in Nature and Society 3 contradicting the choice of G. is proves the claim. Take a longest path P of B(G). Let B 1 be an end block of G, which corresponds to a terminal vertex of P � B 1 c 1 B 2 c 2 B 3 c 3 · · ·, where c 1 be the unique cut vertex of G which belongs to B 1 . By Claim 3, B 2 � K 2 . Next, we consider three possible cases in terms of the order v( is a contradiction.

Case 3:
, which belongs to distinct blocks of G. Let G ′ be the graph obtained from G by contracting B 3 to a vertex v ′ . It can be seen that

Case 3.2:
. . , u s , where u 1 � c 2 and u s � c 3 . Since δ(G) ≥ k, for any i ∈ 1, . . . , s − 1 { }, there are at least k − s + 1 blocks containing u i , each of which is isomorphic to K 2 , as illustrated in Figure 4.

Proof of Theorem 2
Suppose the result is not true and let G be a counterexample of minimum order n. e following fact is clear: (1) G must contain cut vertex.
Since no cubic graph of order ≤ 8 has a cut vertex, n ≥ 10. (2) Moreover, n ≥ 14. If 10 ≤ n ≤ 12, it is not hard to check that b(G) ≤ 3.