The maximum order of partition of the vertex set $V(G)$ into hub sets is called hubtic number of $G$ and denoted by $\xi(G)$. In this paper we determine the hubtic number of some standard graphs. Also we obtain bounds for $\xi(G)$. And we characterize the class of all $(p,q)$ graphs for which $\xi(G)=p$

Shadi Ibrahim Khalaf

Sultan Senan Mahde

Veena Mathad

Opuscula Mathematica

he maximum order of partition of the vertex set V(G) into hub sets is called hubtic number of G and denoted by $\xi(G)$. In this paper we determine the hubtic number of some standard graphs. Also we obtain bounds for $\xi(G)$. And we characterize the class of all (p, q) graphs for which $\xi(G)=p$

Khalaf, S. I.

Mathad, V.

Mahde, S. S.

Biblioteka Nauki - repozytorium artykuÅÃ³w

Hubtic number in graphs

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Opuscula Math. 38, no. 6 (2018), 841–847
https://doi.org/10.7494/OpMath.2018.38.6.841 Opuscula Mathematica
HUBTIC NUMBER IN GRAPHS
Shadi Ibrahim Khalaf, Veena Mathad, and Sultan Senan Mahde
Communicated by Adam Paweł Wojda
Abstract. The maximum order of partition of the vertex set V (G) into hub sets is called
hubtic number of G and denoted by ξ(G). In this paper we determine the hubtic number of
some standard graphs. Also we obtain bounds for ξ(G). And we characterize the class of all
(p, q) graphs for which ξ(G) = p.
Keywords: hubtic number, hub number, partition.
Mathematics Subject Classification: 05C40, 05C99.
1. INTRODUCTION
By a graph G = (V,E) we mean a finite and undirected graph without loops and
multiple edges. A graph G with p vertices and q edges is called a (p, q) graph, the
number p is referred to as the order of a graph G and q is referred to as the size of
a graph G. In general, the degree of a vertex v in a graph G denoted by deg(v) is the
number of edges of G incident with v. Also δ(G) denotes the minimum degree among
the vertices of G [3]. bxc is the greatest integer less than or equal to x. Given any
vertex v ∈ V (G), the graph obtained from G by removing the vertex v and all of its
incident edges is denoted by G− v. In a tree, a leaf is a vertex of degree one. See [3]
for terminology and notations not defined here.
Introduced by Walsh [7], a hub set in a graph G is a set H of vertices in G such
that any two vertices outside H are connected by a path whose all internal vertices lie
in H. The hub number of G, denoted by h(G), is the minimum size of a hub set in G.
A connected hub set in G is a vertex set F such that F is hub set and the subgraph
of G induced by F (denoted G[F ]) is connected. The connected hub number of G,
denoted hc(G), is the minimum size of a connected hub set in G [5]. For more details
on the hub studies we refer to [5–7]. Graphs G1 and G2 have disjoint vertex sets V1
and V2 and edge sets E1 and E2 respectively, their union, G(V,E) = G1 ∪G2 has, as
expected, V = V1∪V2 and E = E1∪E2. Their join is denoted by G1 +G2 and consists
of G1 ∪G2 and all edges joining V1 with V2 [3]. The contraction of a vertex v in G
c© Wydawnictwa AGH, Krakow 2018 841
842 Shadi Ibrahim Khalaf, Veena Mathad, and Sultan Senan Mahde
(denoted by G/v) as being the graph obtained by deleting v and putting a clique on the
(open) neighbourhood of v, (note that this operation does not create multiple edges, if
two neighbours of v are already adjacent, then they remain simply adjacent) [7].
A set D of vertices in a graph G is called dominating set of G if every vertex in
V −D is adjacent to some vertex in D, the minimum cardinality of a dominating set
in G is called the domination number γ(G) of a graph G [4]. In 1977, E.J. Cockayne
and S.T. Hedetniemi introduced the concept of domatic number of graph G and
defined by, a D-partition of G is a partition of V (G) into dominating sets, the domatic
number of G denoted by d(G) is the maximum order of a D-partition of G [1]. Using
the concept of hub set of a graph G and the definition of the domatic number of a
graph G, motivated by this, we introduce the concept of hubtic number of a graph G
as a new parameter of a graph.
A double star Sn,m is the tree obtained from two disjoint stars K1,n−1 and K1,m−1
by connecting their centers [2]. The chromatic number χ(G) of a graph G is the
minimum number of colors required to assign to the vertices of G in such a way that no
two adjacent vertices of G receive the same color [3]. A clique of a graph is a maximal
complete subgraph, and c(G) (resp. C(G)) will denote the smallest (resp. largest)
order of a clique of G [3].
The following results will be useful in the proof of our results.
Theorem 1.1 ([7]). Let ∆(G) denote the maximum degree of G. Then if G is
a connected graph h(G) 6 |V (G)| −∆(G), and the inequality is sharp.
Proposition 1.2 ([1]). For any graph G,
d(G) 6 δ(G) + 1.
Theorem 1.3 ([7]). Let T be a tree with p vertices and l leaves. Then
h(T ) = hc(T ) = p− l.
Theorem 1.4 ([5]). If G is connected, then
hc(G) 6 h(G) + 1.
Theorem 1.5 ([7]). Let S be a subset of V (G). Then G/S is complete if and only if
S is a hub set of G.
2. HUBTIC NUMBER
Definition 2.1. The maximum order of partition of the vertex set V (G) into hub
sets is called hubtic number of G, and denoted by ξ(G). A H-partition of a graph G
is a partition of V (G) into hub sets.
The ihubtic number iξ(G) of a graph G is the maximum order of a partition of
V (G) into hub sets which are independent. In the following observation we determine
the hubtic number of some standard graphs. The proofs of these results are simple
and are omitted.
Hubtic number in graphs 843
Observation 2.2. 1. For any complete graph Kp,
ξ(Kp) = p.
2. For any cycle Cp,
ξ(Cp) =



3, if p = 3,
4, if p = 4,
2, if p = 5, 6,
1, if p > 7.
3. For any path Pp,
ξ(Pp) =



2, if p = 2,
3, if p = 3,
2, if p = 4,
1, if p > 5.
4. For the wheel graph W1,p−1, p > 4,
ξ(W1,p−1) =



4, if p = 4,
5, if p = 5,
3, if p = 6, 7,
2, if p > 8.
5. For the star K1,p−1, p > 4,
ξ(K1,p−1) = 2.
6. For the double star Sn,m,
ξ(Sn,m) = 2.
7. For the complete bipartite graph Kn,m, m,n > 3,
ξ(Kn,m) = min{n,m}.
Observation 2.3. For any graph G, 1 6 ξ(G) 6 p. If h(G) > 0, then
ξ(G) 6
⌊ p
h(G)
⌋
.
Proof. Let H = {H1, H2, H3, . . . ,Hξ(G)} be the hubtic partition of graph G. Clearly,
|Hi| > h(G) for all i = 1, 2, . . . , ξ(G), so
p =
ξ(G)∑
i=1
|Hi| > ξ(G)h(G).
Hence the assertion follows.
844 Shadi Ibrahim Khalaf, Veena Mathad, and Sultan Senan Mahde
Theorem 2.4. Let G be a tree with at least 3 non-leaf vertices. Then ξ(G) = 1.
Proof. Suppose that G is a tree with at least 3 non-leaf vertices. We discuss the
following cases:
Case 1. Let H be a set of all non-leaf vertices. Clearly, any path between two leaves
does not pass through another leaf. So, H is a hub set of G, and by Theorem 1.3,
it is a minimum hub set. Now, suppose that B ⊆ V (G)−H is a hub set of G. Since G
is a tree with at least 3 non-leaf vertices, take any two non-adjacent vertices u, v ∈ H.
Since all vertices in B are leaves, then there is no path between u and v with all
internal vertices in B. This is a contradiction. Hence ξ(G) = 1.
Case 2. Suppose that H is a hub set of G but not containing all non-leaf vertices.
Since G has at least three non-leaf vertices, let {v1, v2, v3} be non-leaf vertices and
v1v3 /∈ E(G), let l1, l3 be leaves adjacent to v1 and v3, respectively. Clearly,
G[{l1, v1, v2, v3, l3}] is a path P5. Since h(P5) = 3, then H contains at least three
vertices from P5. Then any other hub set of G must intersects H since |P5| = 5,
therefore ξ(G) = 1.
Proposition 2.5. For any graph G,
ξ(G) 6 δ(G) + 2.
Proof. Let H = {H1, H2, H3, . . . ,Hξ(G)} be the hubtic partition of graph G. Let
v ∈ Hj for some 1 6 j 6 ξ(G) such that v is not adjacent to any vertex in two hub
sets Hn, Hm and n 6= m 6= j. Then G/Hn is not complete, since v is not adjacent
to any vertex in Hn and in Hm. So by Theorem 1.5, Hn is not a hub set of G,
which is a contradiction, therefore v is adjacent to at least ξ(G) − 2 vertices. Then
ξ(G) 6 δ(G) + 2.
By Propositions 1.2 and 2.5, we get the next result.
Remark 2.6. For any graph G,
ξ(G) + d(G) 6 2δ(G) + 3.
Theorem 2.7. For any non-regular graph G of order p,
ξ(G) + ξ(G) 6 p+ 2.
Proof. By Proposition 2.5, ξ(G) 6 δ(G) + 2, and ξ(G) 6 δ(G) + 2 < ∆(G) + 2. Then
ξ(G) + ξ(G) < δ(G) + ∆(G) + 4 = (p− 1) + 4 = p+ 3.
This yields the desired conclusion.
Theorem 2.8. Let G be a non-regular graph of order p. Then
ξ(G) + h(G) 6 p+ 1.
Hubtic number in graphs 845
Proof. By Theorem 1.1, h(G) 6 p−∆(G). SinceG is non-regular, then h(G) < p−δ(G).
Also from Proposition 2.5 we deduce that ξ(G) 6 δ(G)+2. Now we have ξ(G) 6 δ(G)+2
and h(G) < p− δ(G). It follows that ξ(G) + h(G) < p+ 2. The proof is finished.
By the previous theorem and Theorem 1.4, the next corollary is obvious.
Corollary 2.9. Let G be a connected graph of order p. Then
ξ(G) + hc(G) 6 p+ 2.
Lemma 2.10. Let G be a non-regular graph of order p. Then
h(G) + h(G) 6 p.
Proof. By Theorems 2.7 and 2.8, ξ(G) + h(G) 6 p+ 1 and ξ(G) + h(G) 6 p+ 1. Thus
we get ξ(G) + ξ(G) + h(G) + h(G) 6 p+ 2 + p, which implies the assertion.
Corollary 2.11. Let G be a connected graph of order p. Then
hc(G) + hc(G) 6 p+ 2.
Theorem 2.12. For any graph G of order p, ξ(G) = p if and only if G ∼= Kp, or G
with δ > p− 2.
Proof. Suppose that ξ(G) = p. Then we have p partition of V (G) into hub sets and
every partite set consists of one vertex, we have the following cases:
Case 1. All vertices of G are adjacent, so we can choose every vertex of G as a hub
set, hence G should be complete graph Kp. So δ = p− 1.
Case 2. Any vertex of degree p− 1 is adjacent to all vertices and hence it constitutes
a hub set of G. Since any vertex of degree p− 2 is adjacent to all vertices of G except
one, so every vertex of them must be a hub set for G, hence δ(G) = p − 2, if we
consider any vertex u such that deg(u) < p− 2. In this case {u} is not a hub set for G.
So ξ(G) = p only if the graph G satisfies δ(G) = p− 2. The converse is obvious.
Theorem 2.13. For any (p, q) graph G, we have
ξ(G) >
⌊ p
p− δ(G)
⌋
.
Proof. Let G be a graph. The following cases are considered.
Case 1. G is disconnected graph. Obviously, δ 6 p2 − 1. Then
⌊
p
p−δ(G)
⌋
= 1. Hence
ξ(G) >
⌊
p
p−δ(G)
⌋
.
Case 2. G is a connected graph. If G = Kp, then ξ(G) = p. Let G 6= Kp, we have two
subcases:
846 Shadi Ibrahim Khalaf, Veena Mathad, and Sultan Senan Mahde
Subcase 2.1. δ 6 p2 . If inequality is strict, the result is trivial. Let δ =
p
2 . Then
b pp−δ(G)c = 2. In this case, we can get at least two hub sets for G, thus the result
follows.
Subcase 2.2. δ > p2 , clearly b
p
p−δ(G)c 6
p
2 and ξ(G) >
p
2 , hence we obtain the
assertion.
Proposition 2.14. For any two connected graphs G1 and G2,
ξ(G1 ∪G2) =
{
1, if G1 or G2 is non-complete,
2, if G1 and G2 are complete.
Proof. Let G1, G2 be both complete graphs. Clearly, V (G1) is a hub set for G1 ∪G2
and V (G2) is a hub set of the same graph, thus ξ(G1 ∪G2) = 2. If G1 or G2 is not
complete graph, then any hub set of G1 ∪G2 must contain all of the vertices of G1
and any hub set of G2, therefore ξ(G1 ∪G2) = 1.
Corollary 2.15. If G is disconnected graph with n > 3 components, then ξ(G) = 1.
The inequality c(G) 6 ξ(G) is not true for G ∼= C7. Therefore we define the
following graph.
Definition 2.16. A graph G is said to be inhubable if it has a H-partition in which
every partite set is independent.
Proposition 2.17. Let G be inhubable graph. Then c(G) 6 iξ(G).
Proof. By definition, chromatic number of G is the minimum order of a partition of
V (G) into independent subsets. Clearly that
c(G) 6 C(G) 6 χ(G) 6 iξ(G).
3. CONCLUSION
This paper introduces a new variation in the theory of hub number namely hubtic
number, and just initiates a study on this notion. We list some interesting problems
for further research that we encountered during the course of our investigation.
1. Find a characterization of graphs G for which d(G) = ξ(G).
2. Find a characterization of graphs G for which ξ(G) = 1.
3. Obtain more relations between ξ(G) and the other graph parameters, also obtain
good bounds for ξ(G).
REFERENCES
[1] E.J. Cockayne, S.T. Hedetniemi, Towards a theory of domination in graphs, Networks 7
(1977), 247–261.
[2] J.W. Grossman, F. Harary, M. Klawe, Generalized ramsey theorem for graphs, X: double
stars, Discrete Math. 28 (1979), 247–254.
Hubtic number in graphs 847
[3] F. Harary, Graph Theory, Addison Wesley, Reading Mass, 1969.
[4] T.W. Haynes, S.T. Hedetniemi, P.J. Slater, Fundamentals of Domination in Graphs,
Marcel Dekker, Inc., 1998.
[5] P. Johnson, P. Slater, M. Walsh, The connected hub number and the connected domination
number, Wiley Online Library 3 (58)(2011), 232–237.
[6] Sultan Senan Mahde, Veena Mathad, Some results on the edge hub-integrity of graphs,
Asia Pacific Journal of Mathematics 3 (2) (2016), 173–185.
[7] M. Walsh, The hub number of a graph, Intl. J. Mathematics and Computer Science 1
(2006), 117–124.
Shadi Ibrahim Khalaf
shadikhalaf1989@hotmail.com
Department of Studies in Mathematics
University of Mysore
Manasagangotri
Mysuru – 570 006, India
Veena Mathad
veena_mathad@rediffmail.com
Department of Studies in Mathematics
University of Mysore
Manasagangotri
Mysuru – 570 006, India
Sultan Senan Mahde
sultan.mahde@gmail.com
Department of Studies in Mathematics
University of Mysore
Manasagangotri
Mysuru – 570 006, India
Received: December 5, 2017.
Revised: March 22, 2018.
Accepted: March 29, 2018.


https://doi.org/10.7494/OpMath.2018.38.6.841

Hubtic number in graphs

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Hubtic number in graphs

Abstract

Similar works

Full text

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