Ratio Mathematica Volume 46, 2023 Bounds on fuzzy dominator chromatic number of fuzzy soft bipartite graphs Jahir Hussain Rasheed* Afya Farhana Mohammed Shaik† Abstract An FSG GS(T,V) fuzzy’s soft dominator colouring (FSDC) is a suit- able Fuzzy Soft Colouring (FSC) where every node of a colour group is dominated by a vertex of GS(T,V). In the current work, we charac- terize the sharp bounds for the Fuzzy Dominator Chromatic Number (FDCN) of fuzzy soft bipartite graphs and we present limits on the FDCN of fuzzy soft bipartite graph in terms of the γe(GS(T,V )). Furthermore, we classify fuzzy soft bipartite graphs into three classes based on FDCN. Keywords:Fuzzy Soft Bipartite Graph, Fuzzy dominator chromatic number, Fuzzy soft path, Fuzzy soft cycle, Strong arcs. 2020 AMS subject classifications: 05C72; 05C15. 1 *PG and Research Department of Mathematics, Jamal Mohamed College (Autonomous), Affiliated to Bharathidasan University, Tiruchirappalli - 620 024, Tamil Nadu, India. hssn jhr@yahoo.com. †PG and Research Department of Mathematics, Jamal Mohamed College (Autonomous), Affiliated to Bharathidasan University, Tiruchirappalli - 620 024, Tamil Nadu, India. afya- farhana@gmail.com. 1Received on September 15, 2022. Accepted on December 15, 2022. Published on March 20, 2023. DOI: 10.23755/rm.v46i0.1063. ISSN: 1592-7415. eISSN: 2282-8214. ©R. Jahir Hussain et al. This paper is published under the CC-BY licence agreement. 109 R. Jahir Hussain and M.S. Afya Farhana 1 Introduction Fuzzy soft graphs are a useful mathematical tool for simulating the ambigu- ity of the actual world in view of parameters. Fuzzy soft graphs, which are of- ten utilised in many different disciplines, including decision-making issues, com- bine fuzzy soft sets and the graph model. Scheduling problems are just one of many practical issues involving the allocation of scarce resources for which graph colouring is used as a model. Graph colouring also has an important place in discrete mathematics and combinatorial optimization. To address ambiguous issues in the fields of engineering, social science, eco- nomics, medical research, and environment, Molodstov [1999] created the idea of soft set theory. Fuzzy soft sets, a blend of a fuzzy set and a soft set, were first introduced by P.K. Maji and Biswas [2001]. Thumkara and George introduced the idea of a soft graph in 2014. Rosenfeld first proposed the idea of fuzzy graph theory in 1975. Fuzzy soft graphs were independently introduced in 2015 by Mo- hinta and Samanta as well. Akram and Nawaz [2015] presented fuzzy soft graphs and examined their operations as well as several other graph theoretical ideas. Domination is a fast growing area of graph theory study, and the numerous ways it is used to networks, distributed computers, social networks, and online graphs helps to explain why there is more interest in this subject. The dominator colouring problem in graphs was first described by Gera [2007]. An appropriate colouring of a graph G with the extra characteristic that each vertex in the graph dominates an entire class is known as a dominator colouring of G. The smallest number of colour classes in a graph’s dominator colouring is known as the dom- inator chromatic number. We want to reduce the number of colour categories. In 2007,Gera [2007] explored the dominator chromatic number for the hypercube and more broadly for bipartite graphs. He also presented dominator colorings in bipartite graphs. Chellali and Maffary [2012] studied dominator colorings in some classes of graphs. The idea of fuzzy dominator colouring in fuzzy graphs was created by Hussain and Fathima [2015]. They investigated the fuzzy dominator chromatic number for a number of fuzzy graphs and explored its boundaries. The FDCN of bipartite, middle, and subdivision fuzzy graphs was created by Hussain and Fathima [2015], and established its limits. Domination in fuzzy soft graphs was the subject of study done by Hussain and Hussain [2017]. Fuzzy dominator coloring applied to fuzzy soft graph yields fuzzy soft dom- inator coloring which concentrates on minimizing the number of color classes of FSG. A Soft, Fuzzy Dominator FSG’s colouring should be done in such a way that every node of a colour group is dominated by a vertex of GS(T,V ). The proposed method concentrates on strength of connectedness, strong arc and strong neighbor of fuzzy soft bipartite graph in view of parameters from the existing method. 110 Bounds on fuzzy dominator chromatic number of fuzzy soft bipartite graphs In this paper, we introduced Bounds of Fuzzy Soft Bipartite Graphs with Fuzzy Dominator Chromatic Number. 2 Preliminaries Definition 2.1. Let R be a parameter set and T is a subset of R, Let V = {x1,x2,x3, ·· ·,xn} is a non-empty set. As well (i) α: T → F(V ) (V’s collection of all fuzzy subsets) e a α(e) = αe(say) αe : V → [0,1] (T,α) : Fuzzy Soft Vertex (ii) β: T → F(V× V) (V× V’s collection of all fuzzy subsets) e a β(e) = βe (say) βe : V× V → [0,1] (T,β) : Fuzzy Soft Edge followed by ((T,α),(T,β)) iff βe(x,y) ≤ αe(x) ∧ αe(y) for all e in T and this FSGs are written by GS(T,V ), it is referred to as a Fuzzy Soft Graph (FSG). Additionally, a FSG is a parametrized family unit of fuzzy graphs. Definition 2.2. A Path is intended to be a set of different points x1,x2, · · ·xn in an FSG such that for all e in T and βe (xi−1, xi) > 0, for all i = 1 to n. Definition 2.3. If an FSG GS(T,V ) contains more than 1 smallest arc, ∀ e ∈ T it is referred to as a Fuzzy Soft Cycle. Definition 2.4. A FSG GS(T,V ) is supposed to be a fuzzy soft bipartite if the node set V can be divided into 2 non-empty sets V e1 and V e 2 such that V e 1 and V e 2 are fuzzy independent sets. These sets are called fuzzy bipartition of V, thus each efficient arc of FSG has one end in V e1 and other end in V e 2 , ∀ e ∈ T. Definition 2.5. A fuzzy soft bipartite graph is Complete if for an individual node V e1 , each single node of V e 2 is an efficient neighbour, e ∈ T. 3 Fuzzy soft graphs with fuzzy dominator colouring Definition 3.1. If βe(x,y) = β∞e (x,y), e ∈ T iff the arc (x,y) in FSG is said to be a strong arc, where β∞e (x,y), e ∈ T is the maximum strength of all pathways among x and y. Definition 3.2. A Soft, Fuzzy Dominator a FSG’s colouring should be done in such a way that every node of a colour group is dominated by a vertex of GS(T,V). 111 R. Jahir Hussain and M.S. Afya Farhana Definition 3.3. In a FSDC of FSG, an FDCN is the least number of colour groups, and it is denoted as χefd (G S(T,V)), e ∈ T. 4 Bounds on fuzzy soft bipartite graphs with fuzzy dominator chromatic number Theorem 4.1. Let GS(T,V) be a connected fuzzy soft bipartite graph. Soon after 2 ≤ χefd(G S(T,V )) ≤bn 2 c+1, e ∈ T and these limits are precise. Proof. Take into account a connected fuzzy soft bipartite graph GS(T,V). Ev- ery acceptable fuzzy soft colour must also be present in every fuzzy soft dominator colour, χef(G S(T,V )) = 2, Hussain and Farhana [2020], as a result χefd(G S(T,V )) ≥ 2, e ∈ T. To acquire the upper bound, let V e1 and V e 2 be the 2 bipartite sets of G S(T,V) with the condition |V e1 | ≤ |V e2 |. After that, allocate colours 1,2,...,|V e1 | to the nodes of V e1 and colour |V e1 | +1 to the nodes of V e2 , is a least FSDC. Hence χefd(G S(T,V )) ≤|V e1 | + 1,since |V e1 |≤ b n 2 c, which implies χefd(G S(T,V )) ≤bn 2 c+ 1, e ∈ T.2 Definition 4.1. Consider a FSG GS(T,V) with n nodes. Connect each node of GS(T,V) onto any one of the n isolated nodes, where n is the total number of nodes in the FSG.The resultant graph is called Corona of GS(T,V) and is symbolized as Cor(GS(T,V))= ((T,α1),(T,β1))where αe1(u) = αe(u),u ∈V, e∈T and αe(u) = αe1(u)∈ (0,1], if u is isolated, e∈T. βe1(u,v)= βe(u,v)∈ E, e∈T and βe1(u,v)= α e 1(u) ∧ αe(v), if u ∈ V and v is isolated, e∈T. Remark: 1. Obviously Cor(GS(T,V)) is a FSG if GS(T,V) be a FSG. 2. If GS(T,V) has ’n’ nodes and ’e’ edges then Cor(GS(T,V)) has 2n nodes and e+n edges. 3. Cor(GS(T,V)) has k+n strong arcs if the no. of strong nodes in GS(T,V) is ’k’. Now we discuss about the sharpness of lower bound and upper bound of Theo- rem:4.1. Since the lower bound is sharp, it may be inferred that the entire fuzzy soft bipartite graph’s FDCN is 2. The subsequent theorem proves the sharpness of the upper bound. Theorem 4.2. Having a ’n’-node fuzzy soft path GS(T,V) and n’ is the number of nodes in cor(GS(T,V )). Then χefd(cor(G S(T,V )) = bn ′ 2 c+1, e ∈ T . Proof. Consider a fuzzy soft path having v1,v2,...,vn as nodes. Noticeably GS(T,V) is a fuzzy soft bipartite graph, Hussain and Farhana [2020] and so 112 Bounds on fuzzy dominator chromatic number of fuzzy soft bipartite graphs Cor(GS(T,V )) is also a fuzzy soft bipartite graph. This implies every arc is a strong arc in Cor(GS(T,V )). The number of nodes in GS(T,V) is n, then n’ = 2n nodes in Cor(GS(T,V )). At this instant, we have to colour the nodes of Cor(GS(T,V )). To obtain the least fuzzy soft dominator colouring, every node in minimum dominating set be full of distinctive colour. The nodes in GS(T,V)is strong adjoining to just one node of Cor(GS(T,V )). This implies that every node of GS(T,V) is in a mini- mum dominating set. Therefore γe(Cor(GS(T,V )))=n, e∈T. Color 1,2,...,n is distributed to the nodes of GS(T,V), and colour n + 1 is dis- tributed to the left over nodes. This is the least FSDC of Cor(GS(T,V )). Hence we prove χefd(Cor(G S(T,V ))) = bn ′ 2 c+1, e ∈ T which is the upper bound and it is sharp.2 Now we prove that the values in the middle of 2 and bn ′ 2 c + 1 can be achieved as the FDCN of some fuzzy soft bipartite graphs of order n. Theorem 4.3. Let k be an integer with 2 ≤ k ≤bn 2 c+1, If so, a connected fuzzy soft bipartite graph GS(T,V) with n- nodes and its FDCN is k. Proof. Consider a fuzzy soft path Pek =((T,α1),(T,β1)) with k ≥ 2 nodes and it is assumed as v1,v2,...vk. Now put up a FSG from fuzzy soft path by extending k nodes u1,u2,...,uk in order that uivi ∈ E,(1≤i≤k) and by adding n-2k nodes xj to vk (1≤j≤n-2k). We have n-2k ≥ 1 and k ≥2, it follows that n ≥5, where αe(v1) =αe1(vi),e ∈ T βe(u,v) = βe1(u,v), (u,v)∈ E, e ∈ T βe(ui,vj)= αe(ui) ∧ αe(vj), for every i, e ∈ T. βe(vk, xj)= αe(vk) ∧ αe(xj), for every j, e ∈ T. Clearly, GS(T,V) is a fuzzy soft tree because there are no fuzzy soft cycles of odd length, implying that it is a fuzzy soft bipartite graph, Hussain and Farhana [2020]. At the moment, the sets {v1,v2, ...,vk} and {u1,u2, ...uk−1,vk} have the lowest dominance. As a result,γe(GS(T,V)) = k, e ∈ T . Since γe(GS(T,V)) = k, we require no less than k distinctive colours for least FSDC. Allocate colours 1,2,...,k to v1,v2,...,vk respectively and k+1 more colour to u1,u2,...,uk,x1, x2,...,xn−2k. So each node has k+1 colours offered in GS(T,V) dominate no less than one colour group. At last, the colouring is least FSDC. Alternatively, we allocate k unique colours to the nodes of the set {u1,u2, ...uk−1,vk} then we have a FSDC of ’k’ colours, since we can colour the 113 R. Jahir Hussain and M.S. Afya Farhana remaining nodes using k colours. Thus χefd(G S(T,V )) = k e ∈ T.2 We have a necessary condition for the upper bound. Theorem 4.4. Define GS(T,V) a soft fuzzy bipartite graph with double-edged (U,V). Following this, corona of GS(T,V) is a fuzzy soft bipartite graph whose χefd is b |cor(GS(T,V ))| 2 c+1, e∈ T where |cor(GS(T,V ))| is the number of nodes in cor(GS(T,V )). Proof. Assume a soft fuzzy bipartite graph with bipartition (U,V) and |U| + |V | = number of nodes in GS(T,V). Label GS(T,V)’ be the corona of GS(T,V). Undoubtedly GS(T,V)’ is a fuzzy soft bipartite graph. The points of soft fuzzy bipartite graph allocated with colours 1,2,...|V | and 1+|V | to the left behind nodes, which is a FSDC of GS(T,V)’. Seeing as each node of GS(T,V) dominate itself and the left over nodes dominates its nearest nodes. This colouring is a least FSDC. This implies that each node in minimum dominat- ing set gets an exclusive colour. Hence χefd(cor(G S(T,V ))) = b |cor(G S(T,V ))| 2 c+1, e∈ T . 5 Bounds on fuzzy dominator chromatic number in terms of domination number Theorem 5.1. Define a soft fuzzy bipartite graph GS(T,V). In that case γe(GS(T,V)) ≤ χefd(G S(T,V )) ≤ 2 + γe (GS(T,V)), e∈ T . Proof. Allow GS(T,V) to be a soft fuzzy bipartite graph and c to be the least FSDC with colours 1,2,...,χefd(G S(T,V )). For every colour group of GS(T,V), allow an be a point in the colour group n having 1 ≤ n ≤ χefd(G S(T,V )),e∈ T . We have to prove that S = {an:1 ≤ n ≤χefd(G S(T,V )), e∈ T} is a dominating set. By the definition of FSDC, all nodes of GS(T,V) will dominate all nodes of some colour class. Since S contains a node of each colour group, each node of GS(T,V) dominate some node in S. As a result, S is a fuzzy dominating set, and every least FSDC of GS(T,V) yields a fuzzy dominating set of GS(T,V). Hence γe(GS(T,V)) ≤ χefd(G S(T,V )), e∈ T . We must now validate the upper bound. Because the FCN of fuzzy soft bipar- tite graph is 2, Hussain and Farhana [2020], we can colour the nodes of GS(T,V) using 2 colours 1 and 2. Allot colours 3,4,...,γe(GS(T,V))+2 to the nodes of S and offer colours 1 and 2 to the lasting nodes of GS(T,V) such that 2 strong adjoining 114 Bounds on fuzzy dominator chromatic number of fuzzy soft bipartite graphs nodes be given dissimilar colours. This colouring fall out in fuzzy soft dominator colouring since it is an appropriate fuzzy soft colouring and each node in GS(T,V) dominate all nodes of at least one colour group. Hence χefd(G S(T,V )) ≤ 2 + γe (GS(T,V)), e∈ T .2 Observation: Fuzzy Soft Bipartite Graphs GS(T,V) on the basis of the limits, may be divided into three categories on χefd(G S(T,V )). 1. GS(T,V) is of CLASS 0 if χefd(G S(T,V )) = γe(GS(T,V)), e∈ T . 2. GS(T,V) is of CLASS 1 if χefd(G S(T,V )) = γe(GS(T,V)) + 1, e∈ T . 3. GS(T,V) is of CLASS 2 if χefd(G S(T,V )) = γe(GS(T,V)) + 2, e∈ T . Example:CLASS 0 Consider a fuzzy soft path of k ≥ 2 nodes. Now form a FSG as in Theorem:4.3. Then FSG is of CLASS 0. Example:CLASS 1 Consider a complete fuzzy soft bipartite graph K1,n. The FDCN of K1,n is two and its γe(GS(T,V)) is one, therefore χefd(G S(T,V )) = γe(GS(T,V)) + 1. Hence GS(T,V) is of CLASS 1. Example:CLASS 2 The fuzzy soft cycle of length n≥ 5 has FSDC dn 3 e+2 and its γe(GS(T,V)) is dn 3 e, therefore χefd(G S(T,V )) = γe(GS(T,V)) + 2. Hence GS(T,V) is of CLASS 2. 6 Conclusions In this work, we characterized the bounds on FDCN of fuzzy soft bipartite graphs and also presented bounds in respect of γe(GS(T,V)). Additionally, based on the fuzzy dominator chromatic number (FDCN) that was found, the fuzzy soft bipartite graphs are divided into three categories as fuzzy soft path in CLASS 0, complete fuzzy soft bipartite graph in CLASS 1 and fuzzy soft cycle in CLASS 2. We suggest this study on a few distinct classes of fuzzy soft graphs. References M. Akram and S. Nawaz. 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