Discrete Mathematics 308 (2008) 1330 – 1333
www.elsevier.com/locate/disc
Note
On spanning connected graphs
Cheng-Kuan Lin
a,∗, Hua-Min Huang
b, Jimmy J.M. Tan
a, Lih-Hsing Hsu
caDepartment of Computer Science, National Chiao Tung University, Hsinchu, Taiwan 30010, ROC bDepartment of Mathematics, National Central University, Chungli, Taiwan 32001, ROC
cDepartment of Computer Science and Information Engineering, Providence University, Taichung, Taiwan 43301, ROC Received 24 November 2004; received in revised form 23 March 2007; accepted 26 March 2007
Available online 7 April 2007
Abstract
A k-container C(u, v) of G between u and v is a set of k internally disjoint paths between u and v. A k-container C(u, v) of G is a k∗-container if the set of the vertices of all the paths in C(u, v) contains all the vertices of G. A graph G is k∗-connected if there exists a k∗-container between any two distinct vertices. Therefore, a graph is 1∗-connected (respectively, 2∗-connected) if and only if it is hamiltonian connected (respectively, hamiltonian). In this paper, a classical theorem of Ore, providing sufficient conditional for a graph to be hamiltonian (respectively, hamiltonian connected), is generalized to k∗-connected graphs.
© 2007 Published by Elsevier B.V.
Keywords: Hamiltonian connected; Hamiltonian; Ore Theorem; Menger Theorem
1. Introduction and definitions
For the graph definition and notation we follow[3]. G = (V , E) is a graph if V is a finite set and E is a subset of {(u, v) | (u, v) is an unordered pair of V }. We say that V is the vertex set and E is the edge set. We use n(G) to denote |V |. A graph H is called a subgraph of G if V (H) ⊆ V (G) and E(H ) ⊆ E(G). The induced subgraph G[H] is a subgraph of G where V (G[H ]) = V (H ) and E(G[H ]) = {(u, v) | (u, v) ∈ E(G) and u, v ∈ V (H )}. Two vertices u and v are adjacent if (u, v) is an edge of G. Let v be a vertex of G and H be a subgraph of G. The neighborhood of u respective to H, denoted by NH(u), is {v ∈ V (H ) | (u, v) ∈ E(G)}. The degree dH(u) of a vertex u respective
to H is the number of edges between u and V (H ). The minimum degree of G, written (G), is min{dG(x) | x ∈ V }.
A path is a sequence of vertices represented byv0, v1, . . . , vk with no repeated vertex, and (vi, vi+1) is an edge of G
for all 0i k − 1. We also write the path v0, v1, . . . , vk as v0, . . . , vi, Q, vj, . . . , vk, where Q is a path form vi
to vj. A path is a hamiltonian path if it contains all the vertices of G. A graph G is hamiltonian connected if, for any
two distinct vertices of G, there exists a hamiltonian path joining those two vertices. A cycle is a path with at least three vertices such that the first vertex is the same as the last one. A hamiltonian cycle of G is a cycle that traverses every vertex of G. A graph is hamiltonian if it has a hamiltonian cycle. We use G ∪ H to denote the disjoint union of graph G and graph H. Moreover, we use G ∨ H to denote the graph obtained from G ∪ H by joining all the edges with one vertex in G and the other vertex in H. Let u and v be two nonadjacent vertices of G, we use G + uv to denote the graph obtained from G by adding the edge (u, v).
∗Corresponding author.
E-mail address:cklin@cs.nctu.edu.tw(C.-K. Lin).
0012-365X/$ - see front matter © 2007 Published by Elsevier B.V. doi:10.1016/j.disc.2007.03.072
C.-K. Lin et al. / Discrete Mathematics 308 (2008) 1330 – 1333 1331
A k-container C(u, v) of G between u and v is a set of k internally disjoint paths between u and v. In other words,
C(u, v) consists of paths P1, P2, . . . , Pksuch that E(Pi) ∩ E(Pj) = ∅ and V (Pi) ∩ V (Pj) = {u, v} for 1i = j k.
The concept of container is proposed by Hsu[5]to evaluate the performance of communication of an interconnection network. The connectivity of G,(G), is the minimum number of vertices whose removal leaves the remaining graph disconnected or trivial. It follows from Menger’s Theorem[7] that there is a k-container between any two distinct vertices of G if and only if G is k-connected.
In this paper, we are interested in a special type of container. A k-container C(u, v) of G is a k∗-container if the set of the vertices of all the paths in C(u, v) contains all the vertices of G. A graph G is k∗-connected if there exists a k∗ -container between any two distinct vertices. A 1∗-connected graph except K1and K2is 2∗-connected. A 1∗-connected
graph is actually a hamiltonian connected graph. Moreover, a 2∗-connected graph is a hamiltonian graph. Thus, the concept of k∗-connected graph is a hybrid concept of connectivity and hamiltonicity. The study of k∗-connected graph is motivated by the globally 3∗-connected graphs proposed by Albert et al.[1]. A globally 3∗-connected graph is a cubic graph that is w∗-connected for all 1w 3. Recently, Lin et al.[6]proved that the pancake graph Pnis w∗-connected
for any w with 1w n − 1 if and only if n = 3. Thus, we defined the spanning connectivity ∗(G) of a graph G to be the largest integer k such that G is w∗-connected for all 1wk if G is 1∗-connected graph and undefined otherwise. A graph G is super spanning connected if∗(G) = (G). The complete graph Knis super spanning connected, and
the pancake graph Pnis super spanning connected if and only if n = 3.
Let k be a positive integer. In this paper, we have the following results. If there exist two nonadjacent vertices u and v with dG(u) + dG(v)n(G) + k then G is (k + 2)∗-connected if and only if G + uv is (k + 2)∗-connected. Moreover,
if there exist two nonadjacent vertices u and v with dG(u) + dG(v)n(G) + k, then G is i∗-connected if and only if
G + uv is i∗-connected for 1i k + 2. Assume that dG(u) + dG(v)n + k for all nonadjacent vertices u and v, then
G is r∗-connected for every r ∈ {1, 2, . . . , k + 2}.
2. Sufficient condition for spanning connected graphs
Ore[8,9], and Bondy and Chvátal[2]proved the following theorem:
Theorem 1 (Bondy and Chvátal [2], Ore [8,9]). Assume that there exist two nonadjacent vertices u and v with
dG(u)+dG(v)n(G) then G is 2∗-connected if and only if G+uv is 2∗-connected. Moreover, dG(u)+dG(v)n(G)+1
then G is 1∗-connected if and only if G + uv is 1∗-connected.
Lemma 1. Let k be a positive integer. Suppose that there exist two nonadjacent vertices u and v with dG(u) +
dG(v)n(G) + k. Then, for any two distinct vertices x and y, G has a (k + 2)∗-container between x and y if and only
if G + uv has a (k + 2)∗-container between x and y.
Proof. If G has a (k + 2)∗-container between x and y, then clearly G + uv has a (k + 2)∗-container between x and y. For
the other direction, let C(x, y) = {P1, P2, . . . , Pk+2} be a (k + 2)∗-container of G + uv between x and y. Suppose that
the edge (u, v) /∈ C(x, y). Then C(x, y) forms a desired (k +2)∗-container of G. Thus, we suppose that (u, v) ∈ P1. We
write P1asx, H1, u, v, H2, yand write Piasx, Pi , y for 2 i k + 2. (Note that l(H1) = 0 if x = u, and l(H2) = 0
if y = v.) We set Ci= x, Pi , y, H2−1, v, u, H1−1, x for 2 i k + 2.
Case 1: dG[Ci](u) + dG[Ci](v)n(Ci) for some 2 i k + 2. Without loss of generality, we may assume that dG[C2](u) + dG[C2](v)n(C2). By Theorem 1, there is a hamiltonian cycle C of the induced subgraph G[C2]. Let C =x, R1, y, R2, x. We set Q1=x, R1, y, Q2=x, R2−1, y, and Qi=Pifor 3i k+2. Then {Q1, Q2, . . . , Qk+2}
forms a (k + 2)∗-container of G between x and y.
Case 2: dG[Ci](u) + dG[Ci](v)n(Ci) − 1 for all 2 i k + 2. Since k+2 i=2 (dG[Ci](u) + dG[Ci](v)) = k+2 i=2 (dG[P i](u) + dG[P1](u) + dG[Pi ](v) + dG[P1](v)) =k+2 i=2 (dG[P i](u) + dG[Pi ](v)) + (k + 1)(dG[P1](u) + dG[P1](v))
1332 C.-K. Lin et al. / Discrete Mathematics 308 (2008) 1330 – 1333 x Qk-2 y Qk Qk-1 Q2 Q1 u zj zj+1 v
Fig. 1. Illustration for case 2 of Lemma 1.
= dG(u) + dG(v) + k(dG[P1](u) + dG[P1](v)) n(G) + k + k(dG[P1](u) + dG[P1](v)) and k+2 i=2 (n(Ci) − 1) = k+2 i=2 (n(Pi ) + n(P1)) − (k + 1) =k+2 i=2 n(Pi ) + (k + 1)(n(P1)) − (k + 1) = n(G) + k(n(P1)) − (k + 1),
n(G) + k + k(dG[P1](u) + dG[P1](v))n(G) + k(n(P1)) − (k + 1). Therefore, dG[P1](u) + dG[P1](v)n(P1) − 2.
We claim that dG[Pi ](u)+dG[Pi ](v)n(P
i)+2 for some 2 i k+2. Suppose that dG[Pi ](u)+dG[Pi ](v)n(P
i)+1
for all 2i k + 2. Then
dG(u) + dG(v) = k+2 i=2 (dG[P i](u) + dG[Pi ](v)) + (dG[P1](u) + dG[P1](v)) k+2 i=2 (n(Pi ) + 1) + n(P1) − 2 = n(G) + k − 1.
This contradicts the fact that dG(u) + dG(v)n + k.
Without loss of generality, we may assume that dG[P2 ](u) + dG[P2 ](v)n(P
2) + 2. Obviously, n(P2 )2. We write
P2= x, z1, z2, . . . , zr, y. Then, there exists j ∈ {1, 2, . . . , r − 1} such that (zj, v) ∈ E(G) and (zj +1, u) ∈
E(G). For otherwise, dG[P
2](u) + dG[P2 ](v)r + r − (r − 1) = r + 1 = n(P
2) + 1, giving a contradiction. We
set Q1= x, z1, z2, . . . , zj, v, H2, y, Q2= x, H1, u, zj +1, zj +2, . . . , zr, y, and Qi = Pi for 3i k + 2. Then
{Q1, Q2, . . . , Qk+2} forms a k∗-container of G between x and y. SeeFig. 1for an illustration. With Lemma 1, we have the following theorem:
Theorem 2. Assume that k is any positive integer and there exist two nonadjacent vertices u and v with dG(u) +
dG(v)n(G) + k. Then G is (k + 2)∗-connected if and only if G + uv is (k + 2)∗-connected. Moreover, G is i∗
C.-K. Lin et al. / Discrete Mathematics 308 (2008) 1330 – 1333 1333
Theorem 3 (Ore[9]). Assume that dG(u) + dG(v)n(G) + 1 for all nonadjacent vertices u and v of G. Then G is
1∗-connected.
Theorem 4. Let k be a positive integer. Assume that dG(u) + dG(v)n(G) + k for all nonadjacent vertices u and v
of G, then G is r∗-connected for every 1r k + 2.
Proof. By Theorem 3, G is 1∗-connected and 2∗-connected. Let x and y be two distinct vertices in G. Suppose there
exists an r∗-container{P1, P2, . . . , Pr} of G between x and y for some 2r k + 1. We only need to construct an
(r +1)∗-container of G between x and y. We have dG(y)k +2, for otherwise let w /∈ NG(y)then dG(y)+dG(w)(k +
1) + (n − 2) = n + k − 1, which is a contradiction. We can choose a vertex u in NG(y) −{x} such that (u, y) /∈ E(Pi) for
all 1i r. Without loss of generality, assume that u ∈ Prand we write Pr asx, H1, u, v, H2, y. We set Qi= Pifor
1i r − 1, Qr = x, H1, u, y, and Qr+1= x, v, H2, y. Suppose that (x, v) ∈ E(G). Then {Q1, Q2, . . . , Qr+1}
forms an (r + 1)∗-container of G between x and y. Suppose that (x, v) /∈ E(G). Then, {Q1, Q2, . . . , Qr+1} forms
an (r + 1)∗-container of G + xu between x and y. By Lemma 1, there exists an (r + 1)∗-container of G between x and y.
We give an example to show that the above result may not hold for r = k + 3. Therefore, our result is optimal. Let Knbe a complete graph with n vertices. We set G = (K1∪ Kb) ∨ Kawhere a 3 and b 2. Obviously, (G) = a and
dG(u) + dG(v)2a + b − 1 for any two distinct vertices u and v. Thus, G is not r∗-connected for any r > a.
Dirac[4]proved that any graph G with at least three vertices and(G)n(G)/2 is 2∗-connected. Any graph G with at last four vertices and(G)n(G)/2 + 1 is 1∗-connected. Obviously, if G is a complete graph then it is super spanning connected. Thus, we consider incomplete graphs.
Theorem 5. Assume that G is a graph with n(G)/2+1(G)n(G)−2. Then G is r∗-connected for 1r 2(G)−
n(G) + 2.
Proof. Since n(G)/2 + 1(G)n(G) − 2, n(G)6. Let k be a positive integer and m3. Suppose that n(G) = 2m
and(G) = m + k for some m3 and 1k m − 2. Then dG(u) + dG(v)2(G) = 2m + 2k. By Theorem 4, G is
r∗-connected for 1r 2k + 2. Suppose that n(G) = 2m + 1 and (G) = m + 1 + k for some m3 and 1k m − 2. We have dG(u) + dG(v)2(G) = 2m + 2 + 2k. By Theorem 4, G is r∗-connected for 1r 2k + 3.
Acknowledgements
The authors are grateful to the referees for their thorough reviews of the paper and many helpful suggestions.
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