On a Conjecture about the Saturation Number of Corona Product of Graphs

Let G (VG, EG) be a simple and connected graph. A set M⊆EG is called a matching if no two edges of M have a common endpoint. A matchingM is maximal if it cannot be extended to a larger matching in G. e smallest size of a maximal matching is called the saturation number of G. In this paper, we conrm a conjecture of Alikhani and Soltani about the saturation number of corona product of graphs. We also present the exact value of s(G ∘H) where H is a randomly matchable graph.


Introduction
All graphs considered in this paper are connected and simple; that is, they do not have loops and multiple edges [1][2][3]. For notation and graph theory terminology, we ingeneral follow [11,12,15].
Let G (V G , E G ) be a graph. A collection of edges M G ⊆E G is called a matching of G if no two edges of M G are adjacent. e vertices incident to the edges of a matching M G are said to be saturated by M G (or M G -saturated); the others are said to be unsaturated (or M G -unsaturated). A matching whose edges meet all vertices of G is called a perfect matching of G. If there does not exist a matching M G ′ in G such that |M G | < |M G ′ |, then M G is called a maximum matching of G. A matching M G is maximal if it cannot be extended to a larger matching in G. e cardinality of any maximum matching, ](G), and the cardinality of any smallest maximal matching in G, s(G), are called the matching number and the saturation number of G, respectively.
If any maximal matching in G is also perfect (i.e., if s(G) |V G |/2), then G is called randomly matchable.
Smallest maximal matchings have a wide range of applications in real-world problems. For example, application of smallest maximal matchings related to a telephone switching network was presented in [4]. Finding a smallest maximal matching is NP-hard even for especial family of graphs (such as planar graphs), see [4][5][6]. Also, one can nd some bounds for this invariant in [7][8][9][10]. See [10,12,13] for more details on this topic. See [11][12][13] Recently, Alikhani and Soltani presented the following conjecture about the saturation number of corona product of graphs.
Conjecture. [14] Let G and H be two graphs and |V G | n. en, where ](G) is the size of a maximum matching M G of the graph G and l is the number of M-unsaturated vertices of G.
In this paper, we con rm this conjecture. We also present some more e cient results on the saturation number of corona product of graphs.
For two graphs, G (V G , E G ) and H (V H , E H ). e corona product of G and H, denoted by G ∘ H, is obtained from one copy of G and |V G | copies of H by joining each vertex of the i th copy of H, i ∈ 1, . . . , |V G | , to the i th vertex of G, cf. [15]. In the following, for g ∈ V G , H g shows the copy of H in G ∘ H corresponding to g.

Main Results
e rst result of this section is the proof of the conjecture mentioned in the previous section.

ns(H) ≤ s(G ∘ H) ≤ ns(H) + ](G) + l,
where ](G) is the size of a maximum matching M G of the graph G and l is the number of M-unsaturated vertices of G.
Proof. First, we prove the upper bound. Let M G be a maximum matching of G, and M H be a maximal matching in H that |M H | � s (H). Also, suppose that vertices g 1 , . . . , g l are M-unsaturated vertices of G. ere are two cases for H. where where h i j is the copy of h j in H i corresponding to g i . Easily one can check that M is a maximal matching in G ∘ H. erefore, erefore, ns(H) ≤ s(G ∘ H). e next theorem gives the exact value of s(G ∘ H) for some family of graphs.

Data Availability
No data were used to support this study.   Journal of Mathematics