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

Tavakoli, Mostafa

doi:https://doi.org/10.1155/2022/3375246

Journal of Mathematics

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Abstract Introduction Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 3375246 | https://doi.org/10.1155/2022/3375246

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

Mostafa Tavakoli¹

Academic Editor: Shaofang Hong

Received07 Jun 2022

Accepted20 Aug 2022

Published28 Sept 2022

Abstract

Let be a simple and connected graph. A set is called a matching if no two edges of have a common endpoint. A matching is maximal if it cannot be extended to a larger matching in . The smallest size of a maximal matching is called the saturation number of . In this paper, we confirm a conjecture of Alikhani and Soltani about the saturation number of corona product of graphs. We also present the exact value of where is a randomly matchable graph.

1. Introduction

All graphs considered in this paper are connected and simple; that is, they do not have loops and multiple edges [1–3]. For notation and graph theory terminology, we ingeneral follow [11, 12, 15].

Let be a graph. A collection of edges is called a matching of if no two edges of are adjacent. The vertices incident to the edges of a matching are said to be saturated by (or -saturated); the others are said to be unsaturated (or -unsaturated). A matching whose edges meet all vertices of is called a perfect matching of . If there does not exist a matching in such that , then is called a maximum matching of . A matching is maximal if it cannot be extended to a larger matching in . The cardinality of any maximum matching, , and the cardinality of any smallest maximal matching in , , are called the matching number and the saturation number of , respectively.

If any maximal matching in is also perfect (i.e., if ), then 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 [4–6]. Also, one can find some bounds for this invariant in [7–10]. See [10, 12,13] for more details on this topic. See [11–13] Recently, Alikhani and Soltani presented the following conjecture about the saturation number of corona product of graphs.

Conjecture. [14] Let and be two graphs and . Then,where is the size of a maximum matching of the graph and is the number of -unsaturated vertices of .

In this paper, we confirm this conjecture. We also present some more efficient results on the saturation number of corona product of graphs.

For two graphs, and . The corona product of and , denoted by , is obtained from one copy of and copies of by joining each vertex of the copy of , , to the vertex of , cf. [15]. In the following, for , shows the copy of in corresponding to .

2. Main Results

The first result of this section is the proof of the conjecture mentioned in the previous section.

Theorem 1. Let and be two graphs and . Then,where is the size of a maximum matching of the graph and is the number of -unsaturated vertices of .

Proof. First, we prove the upper bound. Let be a maximum matching of , and be a maximal matching in that . Also, suppose that vertices are -unsaturated vertices of . There are two cases for . Case 1. is a randomly matchable. Suppose that is a maximal matching in that . Set where is the copy of , , in . Clearly, is a maximal matching in . Thus, Case 2. is not a randomly matchable. Thus, is not a perfect matching. Suppose that is a -unsaturated of . Set where is the copy of in corresponding to . Easily one can check that is a maximal matching in . Therefore, Now, we prove the lower bound. Let be a maximal matching of . We consider two below cases for . Case 1. does not have any edges so that has one end in and one end in a copy of . Hence, , and consequently, . Case 2. Suppose be all edges of such that and . Then, for each , we have . Also, for each , we have . On the other hand,Therefore, .
The next theorem gives the exact value of for some family of graphs.

Theorem 2. Let be a graph of order . If is a randomly matchable graph, then

Proof. By Theorem 1, we have . Then, it is sufficient to prove that . Suppose is the copy of in corresponding to . Let is a maximal matching of , and is the copy of , , in . Assume that and . Set(For more illustration, see Figure 1 which is . Suppose . Consider the maximal matching . Since is a randomly matchable graph, then ). According to this fact that is a randomly matchable graph, then is a maximal matching in . Thus,On the other hand, . Therefore, .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright

Copyright © 2022 Mostafa Tavakoli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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