On Domatic and Total Domatic Numbers of Product Graphs

On Domatic and Total Domatic Numbers of Product Graphs P. Francis1 and Deepak Rajendraprasad2 Department of Computer Science, Indian Institute of Technology Palakkad, India. 1: [email protected] 2: [email protected] Abstract A domatic (total domatic) k-coloring of a graph G is an assignment of k colors to the vertices of G such that each vertex contains vertices of all k colors in its closed neighborhood (neighborhood). The domatic (total domatic) number of G, denoted d(G) (dt(G)), is the maximum k for which G has a domatic (total domatic) k-coloring. In this paper, we show that for two non-trivial graphs G and H, the domatic and total domatic numbers of their Cartesian product G H is bounded above by max V (G) , V (H) {| | | |} and below by max d(G), d(H) . Both these bounds are tight for an infinite family of { } graphs. Further, we show that if H is bipartite, then dt(G H) is bounded below by 2min dt(G), dt(H) and d(G H) is bounded below by 2min d(G), dt(H) . These { } { } bounds give easy proofs for many of the known bounds on the domatic and total domatic numbers of hypercubes [8, 31] and the domination and total domination numbers of hypercubes [16, 22] and also give new bounds for Hamming graphs. We also obtain the domatic (total domatic) number and domination (total domination) number of n- n dimensional torus Ck with some suitable conditions to each ki, which turns out to i=1 i be a generalization of a result due to Gravier [13] and give easy proof of a result due to Klavžar and Seifter [23]. Key Words: Cartesian product, Domatic number, Hamming graph, Injective coloring, Torus, Total domatic number. arXiv:2103.10713v1 [math.CO] 19 Mar 2021 2010 AMS Subject Classification: 05C15, 05C69. 1 Introduction All graphs considered in this paper are finite, simple, undirected and do not contain an iso- lated vertex. Let Pn,Cn and Kn respectively denote the path, the cycle and the complete graph on n vertices. Let δ(G) and ∆(G) denote the minimum and maximum degree of a graph G respectively. The neighborhood N(x) of a vertex x is u: ux E(G) and its { ∈ } closed neighborhood N[x] is N(x) x . For S V (G), let S denote the subgraph ∪{ } ⊆ h i induced by S in G. For any graph G, let G denote the complement of G. Let [n] be the 1 set of consecutive integers 1, 2,...,n and Zn be the set of congruence classes of integers { } modulo n. The Cartesian product of two graphs G and H, denoted G H, is a graph whose vertex set is V (G) V (H) = (x, y): x V (G) and y V (H) and two vertices (x ,y ) and × { ∈ ∈ } 1 1 (x ,y ) of G H are adjacent if and only if either x = x and y y E(H) or y = y 2 2 1 2 1 2 ∈ 1 2 and x x E(G). For any vertex u V (G), u V (H) is isomorphic to H. It is 1 2 ∈ ∈ h{ } × i called the H-layer of u and is denoted by Hu. For any vertex v V (H), V (G) v is ∈ h d ×{ }i isomorphic to G called the G-layer of v and is denoted by Gv. For d 2, let Gi denotes ≥ i=1 d G1 G2 Gd. For ri 3, we call Cri a d-dimensional torus. For n 1, q 2, ··· ≥ i=1 ≥ ≥ n the Hamming graph Hn,q is Kq. The special case Hn,2 is a hypercube of dimension n, i=1 denoted as Qn. A domatic (total domatic) k-coloring of a graph G is an assignment of k colors to the vertices of G such that each vertex contains vertices of all k colors in its closed neighborhood (neighborhood). The domatic (total domatic) number of G, denoted d(G) (dt(G)), is the maximum k for which G has a domatic (total domatic) k-coloring. Let D V (G), if N(D) V (G) D then D is a dominating set of G and if N(D) = ⊂ ⊇ \ V (G) then D is a total dominating set of G. The domination (total domination) number of a graph G is the cardinality of a smallest dominating (total dominating) set of G and is denoted γ(G) (γt(G)). In any domatic (total domatic) coloring of a graph G, each color class is a dominating (total dominating) set of G. Thus the domatic and total domatic numbers can be also seen in the following way. The domatic (total domatic) number of G is the maximum number of classes of a partition of V (G) such that each class is a dominating (total dominating) set of G. There is considerable literature on domination and total domination in graphs. See for instance, [6, 13, 20, 21, 23] and a survey of selected topics by Henning [19]. The concept of domatic number and total domatic number was introduced by Cockayne et al., in [10] and [9] respectively, and investigated furtherin[1,2,5,8,12,18,24,26,32,33]. In [33], Zelinka have shown the existence of graphs with very large minimum degree have a total domatic number 1. Chen et al., [8] and Goddard and Henning [12] have studied total domatic coloring under the names coupon coloring and thoroughly dispersed coloring respectively. The motivation for study of total domatic coloring and its applications were mentioned by Chen et al., in [8]. Further, they showed that every d-regular graph G has dt(G) (1 o(1))d/ log d as d , and the proportion of d-regular graphs G for which ≥ − → ∞ dt(G) (1 + o(1))d/ log d tends to 1 as V (G) . In [12], Goddard and Henning have ≤ | |→∞ shown that the total domatic number of a planar graph cannot exceed 4 and conjectured that every planar triangulation G on four or more vertices has dt(G) at least 2. There are some partial answers to this conjecture by Akbari et al., [1] and Nagy [26]. For a bipartite graph G, Heggernes and Telle [18] shown that deciding whether dt(G) 2 is NP-complete. In ≥ [24], Koivisto et al., shown that it is NP-complete to decide whether dt(G) 3 where G is a ≥ bipartite planar graph of bounded maximum degree. Also, they have shown that if G is split 2 or k-regular graph for k 3, then it is NP-complete to decide whether dt(G) k. ≥ ≥ In [8], Chen et al., mentioned that for any graph G, it would be interesting to determine any relations between dt(G) and dt(G G). More generally, for any graphs G and H, we start to determine the relationship between dt(G), dt(H) and dt(G H). In this direction, we prove that if at least one among G or H is bipartite, then G H has a total domatic coloring with 2 min dt(G),dt(H) colors. As consequences, we show that when n is a { } power of 2, the total domatic number of the hypercubes Qn and Qn+1 is n and the torus n C4r is 2n. Also, for any positive integer d and with some suitable conditions to each i=1 i d ki, we show that the total domatic number and total domination number of the torus Ck i=1 i d is 2d and ( ki)/2d respectively. In addition, we obtain similar bounds and results for the i=1 domatic numberQ and domination number of GH. Also, we prove that the domatic and total domatic numbers of G H is upper bounded by max V (G) , V (H) , and lower bounded {| | | |} by max d(G),d(H) . { } The concept of injective coloring was introduced by Hahn et al., in [15] and further studied in [7, 11, 25]. An injective k-coloring of a graph G is an assignment of k colors to the vertices of G such that any two vertices in the neighborhood of each vertex have distinct colors. The minimum k for which such a coloring exists is the injective chromatic number of G, denoted χi(G). Also, the injective chromatic number of a graph G can be seen in the following way. The common neighbor graph G(2) of G has the same vertex set V (G) and any two vertices u, v are adjacent in G(2) if there is a path of length 2 joining u and v in G. (2) 2 It is noted that χi(G)= χ(G ). The square of a graph G, denoted G , has the same vertex set V (G) and edge set E(G) E(G(2)). ∪ We obtain a lower bound for the domatic and total domatic number of Hn−1,q and Hn,q respectively, when n is a power of 2 and q at least 2. In [8], Chen et al., determined the qk−1 injective chromatic number of Hn,q, where q is a prime power and n = q−1 , for some positive integer k. As a consequence, we obtain a lower bound for the domatic and total domatic numbers of Hn−1,q and Hn,q respectively for some more values of n when q is a prime power. 2 Preliminaries It is easy to see that d(G) δ(G)+1 and dt(G) δ(G) for every graph G. We will call the ≤ ≤ graphs which attain these bounds as domatically full and total domatically full respectively.

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