A fuzzy soft dominator colouring of a fuzzy soft graph $G^S$(T,V) is an appropriate fuzzy soft colouring such that every single vertex of $G^S$(T,V) dominate entire vertex of a colour group. In the present work, we initiate fuzzy dominator colouring on fuzzy soft path, fuzzy soft cycle, complete fuzzy soft graph, complete fuzzy soft bipartite graph and its fuzzy soft dominator chromatic number is presented as well as bounds for fuzzy soft dominator chromatic number on fuzzy soft graph is established

Afya Farhana

Jahir Hussain Rasheed

Hussain Rasheed, Jahir

Farhana, Afya

APAV - Academy of Sciences, Letters, Arts and Technology (E-Journals)

Ratio Mathematica Volume 48, 2023Fuzzy dominator colouring on fuzzy softgraphsJahir Hussain Rasheed*Afya farhana Mohamed Sheik Dawood†AbstractA fuzzy soft dominator colouring of a fuzzy soft graph GS(T,V) isan appropriate fuzzy soft colouring such that every single vertex ofGS(T,V) dominate entire vertex of a colour group. In the presentwork, we initiate fuzzy dominator colouring on fuzzy soft path, fuzzysoft cycle, complete fuzzy soft graph, complete fuzzy soft bipartitegraph and its fuzzy soft dominator chromatic number is presented aswell as bounds for fuzzy soft dominator chromatic number on fuzzysoft graph is established.Keywords: Fuzzy Soft 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 March 23, 2023. Accepted on August 1, 2023. Published on October 14, 2023.DOI: 10.23755/rm.v48i0.1153. ISSN: 1592-7415. eISSN: 2282-8214. ©Jahir Hussain Rasheedand Afya farhana Mohamed Sheik Dawood. This paper is published under the CC-BY licenceagreement.213R. Jahir hussain, M.S. Afya farhana1 IntroductionFuzzy 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 ofmany practical issues involving the allocation of scarce resources for which graphcolouring is used as a model. Graph colouring also has an important place indiscrete mathematics and combinatorial optimization.To address ambiguous issues in the fields of engineering, social science, eco-nomics, medical research, and environment, ? created the idea of soft set theory.Fuzzy soft sets, a blend of a fuzzy set and a soft set, were first introduced by Maji?. Fuzzy soft graphs were independently introduced by ? as well. ? presentedfuzzy soft graphs and examined their operations as well as several other graphtheoretical ideas.Domination is a fast growing area of graph theory study, and the numerousways it is used to networks, distributed computers, social networks, and onlinegraphs helps to explain why there is more interest in this subject. The dominatorcolouring problem in graphs was first described by ?. An appropriate colouringof a graph G with the extra characteristic that each vertex in the graph dominatesan entire class is known as a dominator colouring of G. The smallest number ofcolour classes in a graph’s dominator colouring is known as the dominator chro-matic number. We want to reduce the number of colour categories. ? exploredthe dominator chromatic number for the hypercube and more broadly for bipartitegraphs. He also presented dominator colorings in bipartite graphs.? studied dom-inator colorings in some classes of graphs. The idea of fuzzy dominator colouringin fuzzy graphs was created by ?. They investigated the fuzzy dominator chro-matic number for a number of fuzzy graphs and explored its boundaries. Domi-nation in fuzzy soft graphs was the subject of study done by ?.Fuzzy dominator coloring applied to fuzzy soft graph yields fuzzy soft dominatorcoloring which concentrates on minimizing the number of color classes of FSG. Afuzzy dominator FSG’s colouring should be done in such a way that every node ofa colour group is dominated by a vertex of GS(T,V). The proposed fuzzy soft dom-inator coloring concentrates on strength of connectedness, strong arc and strongneighbour in classes of fuzzy soft graph in view of parameters from the existingmethod.In this paper, we introduce fuzzy soft dominator coloring to fuzzy soft graphsand its fuzzy soft dominator chromatic numbers are determined with bounds.214Fuzzy dominator colouring on fuzzy soft graphs2 PreliminariesDefinition 2.1. Let V = {x1, x2, x3, · · ·, xn} is a non-empty set, R is a parameterset and T ⊆ R. Also let(i) α: T → F (V ) (set of all fuzzy subsets in V)e ⊣ α(e) = αe(say)αe : V → [0, 1](T,α) : Fuzzy Soft Vertex(ii) β: T → F(V× V) (set of all fuzzy subsets in V× V)e ⊣ β(e) = βe (say)βe : V× V → [0, 1](T,β) : Fuzzy Soft EdgeThen ((T,α),(T,β)) is called fuzzy soft graph (FSG) iff βe(x, y) ≤ αe(x) ∧ αe(y)for all e ∈ T and this fuzzy soft graphs are symbolized by GS(T, V ). Additionally,a FSG is a parametrized family unit of fuzzy graphs.Definition 2.2. A series of distinctive points x1, x2, · · ·xn in a FSG is mean to bea Path such that ∀ e ∈ T and βe (xi−1, xi) > 0 ∀ i = 1,2,3,· · ·n.Definition 2.3. A FSG GS(T, V ) is called Fuzzy Soft Cycle if it contain above onelowest arc, ∀ e ∈ T.Definition 2.4. A fuzzy soft graph GS(T,V) is complete if βe(x, y) =αe(x) ∧ αe(y),for every x,y ∈ α∗, e ∈ T.Definition 2.5. A FSG GS(T,V) is supposed to be a fuzzy soft bipartite if the nodeset V can be divided into 2 non-empty sets V e1 and Ve2 such that Ve1 and Ve2 arefuzzy independent sets. These sets are called fuzzy bipartition of V, thus eachefficient arc of FSG has one end in V e1 and other end in Ve2 , ∀ e ∈ T.Definition 2.6. A fuzzy soft bipartite graph is complete if for an individual nodeV e1 , each single node of Ve2 is an efficient neighbour, ∀ e ∈ T.3 Fuzzy Soft Dominator Colouring on Fuzzy SoftGraphsDefinition 3.1. The strength of connectedness in a FSG GS(T,V) joining two nodesx,y is define as the greatest strength of the entire paths among x and y and isindicated by β∞e (x, y), ∀ e ∈ T.215R. Jahir hussain, M.S. Afya farhanaDefinition 3.2. An arc (x,y) in FSG GS(T,V) is said to be a strong arc iff βe(x, y)= β∞e (x,y) e ∈ T and x, y is said to be strong adjacent vertices (or) efficientlyadjoining nodes.Definition 3.3. A k-fuzzy soft vertex colouring of a fuzzy soft graph GS(T,V) is anallotment of k –colours, commonly mean as 1,2,· · · k to the vertices of GS(T,V). Afuzzy soft vertex colouring is stated as appropriate colouring if two strong adja-cent vertices receive distinct colours.Definition 3.4. A fuzzy soft dominator coloring (FSDC) of a FSG GS(T,V) is anappropriate fuzzy soft colouring such that every single vertex of GS(T,V) dominateentire vertex of a colour group.Definition 3.5. A fuzzy dominator chromatic number (FDCN) of a FSG is theleast quantity of colour groups in a fuzzy soft dominator colouring of GS(T,V) andit is indicated as χefd (GS(T,V)), e ∈ T.4 Main ResultsProposition 4.1. If GS(T,V) is a fuzzy soft graph such that GS∗(T, V ) = (V,E)is a soft path of size n ≥ 3, in that case χefd(GS(T, V ))= 1 + ⌈n3 ⌉ for n=3,4 and 5and 2+⌈n3⌉ for n ≥ 6, ∀ e ∈ T .Proof. Consider a fuzzy soft graph GS(T,V) of whom fundamental graph is asoft path of size n.=⇒ All interior points of GS(T,V) dominate themselves and their connect-ing points. Hence, it is clear that the fuzzy soft dominator chromatic number ofGS(T,V) for n =3,4 and 5 is 1+⌈n3⌉.Now we have to prove for the case n ≥ 6 as 2+⌈n3⌉.case (i) n≡ 0 (mod 3)Consider a fuzzy soft graph GS(T,V) with points v1,v2,...,v3k . To obtain thefuzzy soft dominator chromatic number of GS(T,V) we have to use no less than3k uncommon colours. With no loss of generality, pick the mid point of every 3nodes, then we have v2,v5,...,v3k−1 having k = ⌈n3 ⌉ colour groups of GS(T,V).Now colour the resting nodes of GS(T,V). We need two colour groups tocolour the resting nodes v3 and v4 to obtain appropriate fuzzy soft colouring sincev3 and v4 are strong arcs. So allot the colour k+1 to v1,v2,...,v3k and assign colourk+2 to v4,v7,...,v3k−2In the end, we have k+2 =⌈n3⌉+2 colour groups in a least fuzzy soft dominatorcolouring.case (ii) n≡ 2 (mod 3)216Fuzzy dominator colouring on fuzzy soft graphsWe should colour an additional node 3k+2 in this case, because it is notstrong adjoin to any node in the equal colour group. So put on this node in thecolour group v1,v2,...,v3k . Hence we get a least fuzzy soft dominator colouring ofGS(T,V) as k+2 =⌈n3⌉+2.Proposition 4.2. If GS(T,V) is a fuzzy soft cycle of length n, then χefd(GS(T, V ))=⌈n3⌉ for n=4 and ⌈n3⌉+2 for n ≥ 5,∀ e ∈ T .Proof. Let Ce4 = v1v2v3v4v1 be a fuzzy soft cycle of size n = 4, for every e∈ T. If the cycle be even, we have two fuzzy independent sets which are v1,v3 andv2,v4. Then allot colour 1 to the nodes v1,v3 and colour 2 to the nodes v2,v4. Hencewe obtain the least fuzzy soft dominator colouring of Cen with χefd(GS(T, V ))=⌈n3⌉ = ⌈43⌉ = 1.3= 2.Now we examine the case n≥ 5 of Cen.Consider a sub graph P en of Cen, e ∈ T with nodes v1,v2,...,vn with the equalmembership function as in Cen. By Proposition 4.1, we have χefd(Pen) = 2+⌈n3 ⌉,e ∈ T and also there exists a least fuzzy soft dominator colouring such that thepoints v1 and vn are dominated by themselves or by its adjacent nodes v2 andvn−1.From the sub graph P en we get the fuzzy soft cycle Cen by joining an edge v1vn.If the colours of v1vn are alike, say colour m (1≤ m ≤ χefd(P en))) e ∈ T which isone of the repeated colour in fuzzy dominator colouring of P en.Because χefd(GS(T, V )) = 2+⌈n3⌉, e ∈ T there is at least one other colour thatrepeats say n (1≤ n ̸= m ≤ χefd(P en)) allot colour n to v1. Now the colour v2 isa non-consecutive colour since v1 need to dominate a few colour group in P en andso the colour v1 and v2 are distinct. The resting nodes have an appropriate fuzzysoft colouring since it was transmitted from P en. The new fuzzy soft colouring isa fuzzy soft dominator colouring because it was a fuzzy soft dominator colouringof P en and the transform in the additional arc were analyzed.Suppose, if the colours of v1 and vn were uncommon colours in a fuzzy softdominator colouring of P en , later the colouring would not be optimal. And so, atthe most one of the node v1 and vn will have an uncommon colour say vn. Thencertainly, vn−1 has a common colour in an optimal fuzzy soft dominator colour-ing of P en . Then joining an edge v1 vn will maintain the fuzzy soft dominatorcolouring of P en of Cen. Thus χefd(GS(T, V )) = 2+⌈n3⌉ for n ≥ 5, ∀ e ∈ T .Proposition 4.3. If GS(T,V) is a complete fuzzy soft graph then χefd(GS(T, V )) =n, ∀ e ∈ T .Proof. Given that GS(T,V) is complete, then every vertex of GS(T,V) domi-nates itself and its adjacent vertices, since all arcs are strong arcs in complete fuzzysoft graph. Hence every node of GS(T,V) dominates a colour group. ThereforeGS(T,V) consist of n colour groups v1,v2,...,vn, χefd(GS(T, V )) = n, ∀ e ∈ T .217R. Jahir hussain, M.S. Afya farhanaProposition 4.4. If GS(T,V) is a complete fuzzy soft bipartite graph thenχefd(GS(T, V )) = 2, ∀e ∈ T.Proof. Since χefd(GS(T, V )) = 2 , which means the FSG contains 2 colourgroups say V e1 and Ve2 . This implies every point of Ve1 dominates each point ofVe2 ,then we have a bipartition as V e1 and Ve2 respectively, ∀ e ∈ T .Therefore GS(T,V) must be a complete fuzzy soft bipartite graph.4.1 Bounds on Fuzzy Dominator Chromatic Number of FuzzySoft GraphsLet GS(T,V) be a connected fuzzy soft graph of size n≥ 2. Then we need noless than two different colours in a fuzzy soft dominator colouring since there aretwo nodes which are strong adjacent to each other. Futhermore, if each node gainsits unique colour, we also obtain a fuzzy soft dominator colouring.Thus2≤ χefd(GS(T, V )) ≤ n, ∀ e ∈ T .and these bounds are sharp for complete fuzzy soft bipartite graph and completefuzzy soft graph.Proposition 4.5. Let GS(T,V) be a connected fuzzy soft graph. Then χefd(GS(T, V ))= 2, ∀ e ∈ T iff GS(T,V) is a complete fuzzy soft bipartite graph.Proof. Consider a fuzzy soft graph with χefd(GS(T, V )) = 2 (i.e.) the graphcontains 2 colour groups V e1 and Ve2 (say). If |V e1 | = 1 (or)|V e2 | = 1, ∀ e ∈ T .=⇒ GS(T,V) is complete fuzzy soft bipartite graph with bipartition |Xe1 |=1 and|Xe2 |= n-1, since GS(T,V) is connected. Suppose we have |V e1 | ≥ 2 and |V e2 | ≥2,∀e ∈ T. Let v ∈ V e1 . Since V e1 is a fuzzy independent set and |V e1 | ≥ 2, it impliesthat the vertex v is not dominated by any node in V e1 . This results that the vertexv is efficiently adjoin to every point in V e2 . Equivalently, for any random point inV e2 , that point is efficiently adjoin to each point in Ve1 . As a result, GS(T,V) is acomplete fuzzy soft bipartite graph. Converse part follows from the proposition4.4.Proposition 4.6. Let GS(T,V) be a connected fuzzy soft graph. Then χefd(GS(T, V ))= n, ∀ e ∈ T if and only if GS(T,V) is a fuzzy soft graph whose fundamental graphis complete and the arcs are strong arcs.Proof. Consider a connected fuzzy soft graph GS(T,V) with fuzzy soft domi-nator chromatic number n. SupposeGS(T,V) is not a fuzzy soft graph with com-plete graph as an underlying crisp graph with strong arcs. Then there exists atthe minimum of one couple of points u,v such that u and v are not efficiently218Fuzzy dominator colouring on fuzzy soft graphsadjoin. Represent fuzzy soft colouring on GS(T,V) in which u and v taking thesame colour and the rest of the nodes meet with unique colour. Therefore,GS(T,V)receives a fuzzy soft dominator colouring with χefd(GS(T, V )) = n-1 which is acontradictory statement for χefd(GS(T, V )) = n, ∀ e ∈ T . Converse part followsfrom proposition 4.3.Proposition 4.7. A fuzzy soft graph GS(T,V) have a point of efficiently adjoindegree n-1 then χefd(GS(T, V )) = χef (GS(T, V )), ∀ e ∈ T .Proof. Let GS(T,V) be a connected fuzzy soft graph with n points. Supposethat GS(T,V) having a node of strong adjacent degree n-1. Let v1, v2 ... vn bethe points of GS(T,V). With no loss of generality, let vi for some i (i=1,2,... n)be the point of degree n-1 which is strong. Then vi is efficiently adjoin to everypoint of GS(T,V). If c1, c2, c3 ... ck be the colour groups of GS(T,V) such thatχef (GS(T, V )) = k, ∀e ∈ T. Clearly the node vi is in one of the colour group sayci. This is a fuzzy soft colouring which is appropriate and each node of GS(T,V)dominates at least one colour group (ci.) Hence we have a fuzzy soft dominatorcolouring of k colour groups. Therefore χefd(GS(T, V )) = k = χef (GS(T, V )), ∀e ∈ T .Corolary 4.1. If the fuzzy soft domination number of GS(T,V) is equal to 1, thenχefd(GS(T, V )) = χef (GS(T, V )), ∀ e ∈ T .Proposition 4.8. Let GS(T,V) be a connected fuzzy soft graph. Then,max (χef (GS(T, V )),γe(GS(T, V ))) ≤ χefd(GS(T, V )) ≤ χef (GS(T, V )) +γe(GS(T, V )), ∀ e ∈ T .Proof. Let GS(T,V) be a connected fuzzy soft graph, since every fuzzy softdominator colouring must be an appropriate fuzzy soft colouring, we haveχef (GS(T, V ))≤ χefd(GS(T, V )). In addition, let c be a least fuzzy soft dominatorcolouring ofGS(T,V). For every colour group ofGS(T,V) let ui be a point in thegroup i with 1≤ i ≤ χefd(GS(T, V )). Then S = ui:1≤ i ≤ χefd(GS(T, V )) is adominating set which gives γe(GS(T, V )) ≤ |S| = c =χefd(GS(T, V )). Thereforemax (χef (GS(T, V )),γe(GS(T, V ))) ≤ χefd(GS(T, V )), ∀ e ∈ T .For the upper bound, let c be an appropriate fuzzy soft colouring of GS(T,V)with χef (GS(T, V )) colours. Now assign colours χef (GS(T, V )) + 1, χef (GS(T, V ))+2,... χef (GS(T, V )) + γe(GS(T, V )) to the points of least dominating set ofGS(T,V) departing the remaining points coloured already. Clearly this is a fuzzysoft dominator colouring of GS(T,V). Hence χefd(GS(T, V )) ≤ χef (GS(T, V )) +γe(GS(T, V )), ∀ e ∈ T .Corolary 4.2. If GS(T,V) be complete fuzzy soft bipartite graph, then the lowerbound max (χef (GS(T, V ))),γe(GS(T,V)) ≤ χefd(GS(T, V )),∀e ∈ T is sharp.219R. Jahir hussain, M.S. Afya farhanaProof. Let GS(T,V) be complete fuzzy soft bipartite graph, thenmax (χef (GS(T, V )),γe(GS(T, V ))) = max(2,2) = 2 ≤ χefd(GS(T, V ))= 2, ∀ e ∈T .Corolary 4.3. If GS(T,V) be a fuzzy soft path of length n≥6, then the upper boundχefd(GS(T, V )) ≤ χef (GS(T, V )) + γe(GS(T, V )), ∀ e ∈ T is sharp.Proof. Let GS(T,V) be a fuzzy soft path of length n≥ 6, thenχefd(GS(T, V ))=2+⌈n3⌉ ≤ χef (GS(T, V )) + γe(GS(T, V ))=2 + ⌈n3 ⌉, ∀ e ∈ T .5 ConclusionIn this work, we defined fuzzy soft dominator colouring for classes of fuzzysoft graphs and its fuzzy soft dominator chromatic numbers through bounds werestudied. In this study, we analysed the bounds for the connected fuzzy soft graphof size n ≥ 2 is 2 ≤ χefd(GS(T, V )) ≤ n, ∀ e ∈ T which is sharp for completefuzzy soft bipartite graph and complete fuzzy soft graph. And also, the lowerbound for complete fuzzy soft bipartite and the upper bound for fuzzy soft path issharp in terms of domination number. We further extend this study on some moretypes of fuzzy soft graphs.ReferencesChellali and Maffary. Dominator colorings in some classes of graphs. GraphsCombin, 28:97–107, 2012.D.Molodstov. Soft set theory – first results. Computers and Mathematics withApplications, 37:19–31, 1999.P. M. et.al. Fuzzy soft sets. Journal of Fuzzy Mathematics, 9(3):589–602, 2001.R. Gera. On the dominator colorings in bipartite graphs. Proceedings of the 4thInternational Conference on Information Technology:New Generations, pages947–952, 2007a.R. Gera. On the dominator colorings in graph. Graph Theory Notes of New YorkLII, pages 25–30, 2007b.R. J. Hussain and K. K. Fathima. On fuzzy dominator colouring in fuzzy graphs.Applied Mathematical Sciences, 9:1131–1137, 2015.R. J. Hussain and S. Hussain. Domination in fuzzy soft graphs. InternationalJournal of Fuzzy Mathematical Archieve, 14:243–252, 2017.220Fuzzy dominator colouring on fuzzy soft graphsM.Akram and S.Nawaz. On fuzzy soft graphs. Italian journal of pure and appliedmathematics, 34:497–514, 2015.Mohinta and Samanta. An introduction to fuzzy soft graph. Mathematica Morav-ica, 19(2):35–48, 2015.221

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Fuzzy dominator coloring on fuzzy soft graphs

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Fuzzy dominator coloring on fuzzy soft graphs

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