Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 148 https://internationalpubls.com Some Results of Total Domatic Number on Anti Fuzzy Graph R. Muthuraj1*, P. Vijayalakshmi2 and A. Sasireka3 1Research Supervisor & Associate Professor, PG & Research Department of Mathematics, H.H. The Rajah’s College, Pudukkottai – 622 001, Tamilnadu, India. 2 Research Scholar, PG & Research Department of Mathematics, H.H. The Rajah’s College, Pudukkottai – 622 001, Affiliated to Bharathidhasan University, Tiruchirappalli, Tamilnadu, India. Assistant Professor, Department of Mathematics, PSNA College of Engineering and Technology, Dindigul – 624 622, Tamilnadu, India. 3Assistant Professor, Department of Mathematics, PSNA College of Engineering and Technology, Dindigul – 624 622, Tamilnadu, India. e-mail rmr1973@gmail.com Article History: Received: 10-07-2024 Revised: 23-08-2024 Accepted: 06-09-2024 Abstract: Let AG = (N, A, σ, μ) be an anti fuzzy graph. A partition DP = {D1, D2, ….., DK} of N(AG) is referred to as total domatic partition of AG if for each Di is a total dominating set of anti fuzzy graph AG and N(AG)= ⋃Di. The maximum cardinality taken over all maximum number of classes with a minimal total domatic partition of AG is called the total domatic number of AG and it is denoted by 𝑑𝑡(𝐴𝐺). The maximum number of classes with maximum fuzzy cardinality of a partition Di (AG) is called anti fuzzy total domatic number of anti fuzzy graph AG and it is denoted by 𝑑𝑓𝑡(𝐴𝐺). In this paper, we gain some preferred results and limits that referring to the full domatic number on anti fuzzy graph. Keywords: Anti fuzzy graph, Dominating set, Total dominating set, Vertex degree. 1. Introduction The notion of an anti-fuzzy structure on a graph was familiar to Muhaamad Akram [1] owing to the fuzzy relation pioneered by Zadeh [11]. E. J. Cockayne, S. T. Hedetniemi[2] delivered the idea of domatic number of a graph. The idea of a graph's anti domatic number was first developed by Bohdan Zelinka [13]. Domatic number and total domatic number of complete uniform hypergraphs and complete bipartite uniform hypergraphs were computed by Dash, S.P. [3]. The generalities of certain different forms of anti-fuzzy graphs were presented by R. Muthuraj and A. Sasireka [6, 7&8] who also determined the domination parameters on anti-fuzzy graphs. Additionally, they invented the concept of the anti-fuzzy graph's total domination number and established boundaries for it. In this paper, we define the definition of total domatic number and partial total domatic number on anti fuzzy graph AG also extant some general bounds and results that relate the total domatic number of AG. Note The total dominating set D of AG contained each support node in AG. In both N\D and D, it able to dominate many nodes. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 149 https://internationalpubls.com 2. SOME RESULTS OF TOTAL DOMATIC NUMBER ON ANTI FUZZY GRAPH 2.1 Definition Let AG = (N, A, σ, μ) be an anti fuzzy graph. A partition DTP = {TD1, TD2, …., TDK} of N(AG) is called total domatic partition of AG if for each TDi is a total dominating set [TDS] of anti fuzzy graph AG and N(AG) =⋃TDi. The maximum cardinality taken over all maximum number of classes with a minimal total domatic partition of AG is called the total domatic number [TDTN] of AG and it is denoted by 𝑑𝑡(𝐴𝐺). The maximum number of classes with maximum fuzzy cardinality of a partition TDi (AG) is called anti fuzzy total domatic number of anti fuzzy graph and it is denoted by 𝑑𝑓𝑡(𝐴𝐺). 2.2 Example Figure. 1. Anti Fuzzy Graph AG From figure 1, the total dominating sets are TD1 = {u2, u5} = {0.4, 0.6} = 1 TD2 = {u3, u4} = {0.7, 0.3} = 1 TD3 = {u1, u6} = {0.2, 0.5} = 0.7 TDP = {TD1, TD2, TD3} TDT number of anti fuzzy graph AG, 𝑑𝑡(𝐺𝐴) = 3 Anti fuzzy total domatic number of anti fuzzy graph AG, 𝑑𝑓𝑡(𝐴𝐺) = max {1, 1, 0.7} = 1 2.3 Definition Let AG = (N, A, σ, μ) be an anti fuzzy graph. A partition TDP = {TD1, TD2, …., TDK} of N(AG) is called partial total domatic partition of AG if for every TDi is a total dominating set of anti fuzzy graph AG and at the minimum of single node does not in any one of TDi and all TDi’s are minimal total dominating sets. The maximum fuzzy cardinality taken over all maximum number of classes with minimal partial total domatic partition of AG is called the partial total domatic number [PTDTN] of AG and it is denoted by 𝑑𝑝𝑡(𝐴𝐺). The maximum number of classes with maximum fuzzy cardinality of a partition TDi (𝐴𝐺) is called the anti fuzzy partial total domatic number of AG and it is denoted by 𝑑𝑓𝑝𝑡(𝐴𝐺). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 150 https://internationalpubls.com 2.4 Example Figure. 2. Anti Fuzzy Graph AG From figure 2, the total dominating set is TD1 = {u1, u3} = {0.5, 0.7} dfpt (AG) = {0.5, 0.7} = 1.2 For finding TD2, u5 is isolated node. So, it dominates itself and ˂TD2˃ is not a total dominating set. Since, TD2 does not exist. Therefore TDP = {TD1} Partial total domatic number of AG domatic number of AG, 𝑑𝑝𝑡(𝐴𝐺) = 1 Anti fuzzy partial total domatic number of AG domatic number of 𝐴𝐺 , 𝑑𝑓𝑝𝑡 = 1.2 2.5 Theorem Let AG be a finite undirected AFG with n nodes of order ρ, and τ(AG) be the minimum degrees of nodes of AG. Then 𝑑𝑓𝑡(𝐴𝐺) ≥ [ρ/ (ρ- 𝜏𝑓(𝐴𝐺) + 1)] and each total dominating set consists n - τ(AG) + 1 nodes of AG. Proof Let AG be an AFG and 𝐴𝐺̅̅̅̅ is complement of AG. TD is a total dominating set of AG which is a subset of the node set N(AG). For every k ϵ N(AG) there exists a node l which is not adjacent to k in 𝐴𝐺̅̅̅̅ . Suppose the node has degree r in AG, then its degree is n-r-1 in 𝐴𝐺̅̅̅̅ . Therefore, the maximum degrees of 𝐴𝐺̅̅̅̅ is 𝜌 − 𝜏𝑓(𝐴𝐺) − 1. Let TD be a subset of N(AG) having at the minimum of n - τ(AG) + 1 nodes. Then every node k ϵ N(AG) can be contiguous to at most n - τ(AG) + 1 nodes of TD in 𝐴𝐺̅̅̅̅ ; although k ϵ TD, then a node l ϵ TD which is not contiguous to k in 𝐴𝐺̅̅̅̅ and thus is contiguous to k in 𝐴𝐺̅̅̅̅ . Which mean it every subset of N(AG) with at the minimum of n - τ (AG) + 1 nodes is a total dominating set in AG. Consider a partition of N(AG) into a class which having n - τ (AG) + 1 nodes each, with the exception of at most one which would have more nodes. Evidently  such a partition having [n/ (n - τ(AG) + 1)] classes with 𝑑𝑓𝑡(𝐴𝐺) ≥ [ρ/(ρ- τ𝑓(AG) + 1)] this is a total domatic partition. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 151 https://internationalpubls.com Hence 𝑑𝑓𝑡(𝐴𝐺) ≥ [ρ/ (ρ- τ𝑓(AG) + 1)]. 2.6 Theorem AG is an AFG with n nodes, 3≤n≤7 for which τ(AG) = n – 3 and 𝑑𝑓𝑡(𝐴𝐺) = { ≤ [ 𝜌 2 ] ; 𝑓𝑜𝑟 𝑛 = 4,7 ≥ [ 𝜌 2 ] ; 𝑓𝑜𝑟 𝑛 = 6 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡 ; 𝑓𝑜𝑟 𝑛 = 3, 5 . Proof For n = 3, AG is an AFG with three isolated nodes. Since τ(AG) = n – 3. Therefore, total dominating set does not exist. Hence 𝑑𝑓𝑡(𝐴𝐺) = 0. For n = 4, AG is a disconnected anti fuzzy graph with two components with two nodes each. Therefore, there exist one partition of total dominating set. Therefore, 𝑑𝑓𝑡(𝐴𝐺) = [ 𝜌 2 ]. For n = 5, AG is an anti fuzzy cycle. We know that, for any anti fuzzy cycle total domatic partition does not exist. For n = 6, AG is an anti fuzzy wheel with two TDS of at most 𝑛 2 nodes which has at the minimum of 𝜌 2 . Therefore, 𝑑𝑓𝑡(𝐴𝐺) ≥ [ 𝜌 2 ]. For n = 7, AG is an AFG with ∆(𝐴𝐺) = 𝑛 − 2. It forms three total dominating sets with at most 𝑛 2 nodes which has at most [ 𝜌 2 ] each. Therefore, 𝑑𝑓𝑡(𝐴𝐺) ≤ [ 𝜌 2 ]. 2.7 Theorem If AG is an AFG (n=5) with τ(AG) = n – 3 then 𝑑𝑓𝑝𝑡(𝐴𝐺) = [ 𝜌 2 ]. Proof If AG is an AFG with n nodes and τ(AG) = n – 3 then AG has an anti fuzzy cycle. Therefore, there exist one partial total dominating set exist with at most [ 𝜌 2 ]. Hence, 𝑑𝑓𝑝𝑡(𝐴𝐺) = [ 𝜌 2 ]. 2.8 Theorem Let AG be a complete bipartite AFG with ‘n’ nodes. N1 and N2 are node partition of N(AG). Then 𝑑𝑡(𝐴𝐺) = ⌊ 𝑛 2 ⌋. Proof Let AG be any complete bipartite AFG with disjoint node partitioned set N1, N2 and |𝑁1(𝐴𝐺) | = 𝑛, |𝑁2(𝐴𝐺) | = 𝑚 consider n ˂ m (= n+1) There is no edge between the nodes in N1 and Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 152 https://internationalpubls.com also in N2. Let k1 ϵ N1(AG) which dominates all the nodes in N2. Let k1ϵ N2(AG) which dominates all the nodes in N1. Since there exist an edge between k1 and l1. Therefore {k1, l1} forms a minimal TDS of AG. Similarly, {k2, l2}, {k3, l3}, ..., {kn-1, ln-1} forms a minimal TDS of AG. forms a minimal TDS of AG and {kn, ln, ln+1} forms a TDS of AG. Therefore, 𝑑𝑡(𝐴𝐺) = ⌊ 𝑛 2 ⌋. Consider if n = m then {k1, l1}, {k2, l2}, {k3, l3}, …., {kn, l n} are classes of total domatic partition of AG. Therefore, 𝑑𝑡(𝐴𝐺) = ⌊ 𝑛 2 ⌋. 2.9 Proposition For AFG AG, 𝑑𝑓𝑡(𝐴𝐺) ≤ 2𝜌 3 where AG is a not an anti fuzzy cycle. Proof Let AG is an AFG and consider that AG is not an anti fuzzy cycle with n nodes and its order ρ. Since every node in AG has adjacent to at the minimum of two nodes and does not have any pendent node. So, each node of AG dominates at the minimum of two. Therefore, it frames at most three minimal total dominating sets of TDTP of AG. Hence 𝑑𝑓𝑡(𝐴𝐺) ≤ 2𝜌 3 . 2.10 Theorem Let AG be a simple connected AFG and 𝐴𝐺̅̅̅̅ be an anti-complement of AG then 𝑑𝑓𝑡(𝐴𝐺) + 𝑑𝑓𝑡(𝐴𝐺̅̅̅̅ ) ≤ 5𝜌 3 . Proof AG is a simple connected AFG without isolated nodes then 𝐴𝐺̅̅̅̅ does not have any isolated nodes then 𝑑𝑓𝑡(𝐴𝐺) ≤ 2𝜌 3 & 𝑑𝑓𝑡(𝐴𝐺̅̅̅̅ ) ≤ ρ. 𝑑𝑓𝑡(𝐴𝐺) + 𝑑𝑓𝑡(𝐴𝐺̅̅̅̅ ) ≤ 2𝜌 3 + 𝜌 ≤ 5ρ 3 . 2.11 Theorem For any complete uninodal AFG AG with n nodes, 𝑑𝑓𝑡(𝐴𝐺)={ 2σ(𝑢1); 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛 3σ(𝑢1); 𝑖𝑓𝑛 𝑖𝑠 𝑜𝑑𝑑 for all k1∈ N(AG). Proof Consider AG is a complete uninodal AFG and TD is a total domatic partition of AG which has TD1, TD2, …, are its classes. Which yields the classes TD1, TD2, …, TDn/2 are total dominating sets with same cardinality. Let k1ϵ TD1 and has adjacent to n-1 nodes with degree (n-1) k1. l1ϵN(AG) and k1, l1ϵ TD1. Since, k1, l1 are also adjacent and dominates all other nodes in AG. If n is an even number, we get TD1, TD2, …, TDn/2 classes in total domatic partition of AG. Hence, 𝑑𝑓𝑡(𝐴𝐺) = 2σ(𝑘1). If n is odd then AG has TD1, TD2, …, TD𝑛 2 −1 classes have equal number of nodes which forms a TDS classes in total domatic partition of GA. But kn does not belongs to any other classes of total Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 153 https://internationalpubls.com domatic partitions of AG. Therefore, the node kn adding into the total dominating set |TD𝑛 2 | = 2σ(𝑘1) + σ(𝑘𝑛) = 2σ(𝑘1) + σ(𝑘1) {since AG is uninodal anti fuzzy graph} = 3σ(𝑘1) Hence 𝑑𝑓𝑡(𝐴𝐺) = 3σ(𝑘1). 2.12 Theorem If AG is an AF path, then 𝑑𝑓𝑡(𝐴𝐺) ≤ 𝜌 − 𝜏, where τ is minimum degree of AG. Proof Consider AG is an AF path with order ‘ρ’ and has minimum degree τ. Let TD be minimal TDS of AG. Let k and l are the initial and end node of an anti fuzzy path AG. Since, it has the degree as one and the remaining nodes of AG has degree two. Therefore, alternative pair of nodes consist in TDS. Hence, 𝑑𝑓𝑡(𝐴𝐺) ≤ 𝜌 − 𝜏 2.13 Theorem For any two anti fuzzy graphs AG and AH without an isolated node, then the following conditions holds. (i) 𝑑𝑓𝑡(𝐴𝐺 × 𝐴𝐻) ≥ 𝑑(𝐴𝐺) ∨ 𝑑(𝐴𝐻) (ii) 𝑑(𝐴𝐺 × 𝐴𝐻) ≥ 𝑑𝑡(𝐴𝐺 × 𝐴𝐻). Proof (i) Consider AG and AH are anti fuzzy graphs with order ρ1 and ρ2 respectively. Let ρ1≥ ρ2 with n1 ≥ n2 where n1 and n2 are number of nodes of AG and AH. Let TD be a total domatic partition of 𝐴𝐺 × 𝐴𝐻 which having at most n2 classes. Let ρ1, ρ2 be the domatic numbers of AG and AH respectively. If ρ1 ≥ ρ2 then TD1, TD2, …., TDn1 be the domatic partition of N(GA) for 1 ≤ i ≤ n1, any node u1 ϵ TDi and v1 ϵN(HA) then the node (u1, v1) ϵ 𝐴𝐺 × 𝐴𝐻 dominates at most four nodes in N (𝐴𝐺 × 𝐴𝐻). Since HA does not have any isolated node then TD is a total domatic partition of 𝐴𝐺 × 𝐴𝐻 with ρ1, Hence, 𝑑𝑓𝑡(𝐴𝐺 × 𝐴𝐻) ≥ 𝑑(𝐴𝐺) ∨ 𝑑(𝐴𝐻). (ii) Since DP is a domatic partition of AG which have at most n1 classes. Since, a single node can dominate all other nodes in 𝐴𝐺 × 𝐴𝐻. But to form a total dominating set we need at the minimum of two nodes in each domatic partition of 𝐴𝐺 × 𝐴𝐻 . Therefore, 𝑑(𝐴𝐺 × 𝐴𝐻) ≥ 𝑑𝑡(𝐴𝐺 × 𝐴𝐻). 3. CONCLUSION Total and partial total domatic number on an anti fuzzy graph AG, and they are applied to different types of anti-fuzzy graphs to produce bounds. The bounds on them were established by applying the total domatic number concept to the anti-cartesian product of anti-fuzzy graphs such as path, anti-fuzzy cycle, and full anti-fuzzy graph. A few theorems and propositions are produced for the results once they have been analysed. 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