On Domatic and Total Domatic Numbers of Product Graphs

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(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. Regular total domatically full graphs are also called rainbow graphs (see, [27, 29]). We ﬁrst make some easy observations on rainbow graphs. Examples of rainbow graphs include cycles Cn where n 0 (mod 4), Kn,n, Kn K , etc. ≡ 2 Proposition 2.1. Let G be an r-regular total domatically full graph. Every r-total domatic coloring of G, say f : V (G) [r] satisﬁes the following. →

3 |V (G)| (i) Each color class of f has the same size r . ([29, 32]) (ii) Each colorclassof f has an even number of vertices and G contains a perfect matching. (iii) r divides V (G) and r2 divides E(G) . ([29]) | | | | |V (G)| (iv) γt(G)= r .

(v) dt(G)= χi(G)= r.

Proof. (i) Let V and V be two color classes of c and c respectively, where c ,c [r]. 1 2 1 2 1 2 ∈ Each vertex of V1 is adjacent to exactly one vertex of V2 and vice versa and thus V1 = V2 . |V (G)| | | | | Hence f partitioned V (G) into r classes having the same size r (see, [29, 32]). (ii) For any vertex x V (G), there exists exactly one neighbor of x having the color f(x). ∈ Thus each color class Vi of f induces a perfect matching in Vi and hence Vi is even. Also, h i | | G contains a perfect matching which is the union of perfect matchings of all color classes. (iii) Clearly, r divides V (G) and r2 divides E(G) which follows from the fact that E(G) = |V (G)|r |V (G)| | | | | | | 2 and r is even (see, [29]). |V (G)| (iv) Each color class Vi of f is a total dominating set, thus γt(G) . Also, γt(G) ≤ r ≥ |V (G)| = |V (G)| , since any set S of size smaller than |V (G)| can dominate at most S ∆(G) < ∆(G) r ∆(G) | | V (G) vertices. | | (v) Clearly, f is also an injective coloring and the proof follows from χi(G) ∆(G) = ≥ r.

A complete characterization of regular domatically full graphs was done by Zelinka [30].

Examples of regular domatically full graphs include cycles Cn where n 0 (mod 3), Kn, ≡ etc.

Theorem 2.2. [30] An r-regular graph G is domatically full if and only if r +1 divides V (G) and G has an (r + 1)-coloring f such that each color class of f is an independent | | |V (G)| set of size r+1 and the subgraph induced by the vertices of any two color classes of f is a perfect matching.

The domatic coloring of an r-regular graph is closely associated with the proper injective coloring which follows in Corollary 2.3. We say that an injective coloring is proper if no two adjacent vertices get the same color.

Corollary 2.3. Let G be an r-regular graph. G is domatically full if and only if G has a |V (G)| proper (r + 1)-injective coloring. Also, γ(G)= r+1 .

3 Domatic and total domatic numbers of Cartesian products

The union of two disjoint dominating sets of G should be a total dominating set for G and d(G) thus dt(G) d(G) 2dt(G)+1. In any total domatic coloring of G, each 2 ≤ ≤ ≤ j k 4 |V (G) vertex should receive its own color from a neighbor, thus dt(G) and also, d(G) ≤ 2 ≤ 1 V (G) . Hence it follows that, for any two graphs G and H, j2 maxk d(G),d(H) | 1 | 1 1 { } ≤ d(G H) dt(G H) V (G H) = V (G) V (H) and d(G H) 2 ≤ ≤ 2 | | 2 | || | ≤ V (G H) = V (G) V (H) . We improve these bounds given above for any two graphs in | | | || | Theorem 3.1.

Theorem 3.1. For any two graphs G and H without an isolated vertex, we have

max d(G),d(H) dt(G H) d(G H) max V (G) , V (H) . { } ≤ ≤ ≤ {| | | |} Proof. Let G and H be graphs of order m and n respectively. Now, let us prove the upper bound for d(G H). Without loss of generality, let n m. Let us consider the coloring of ≥ G H as ﬁlling the cells of m n grid with colors. For a cell (i, j), 1 i m, 1 j n, × ≤ ≤ ≤ ≤ call the set of cells in the ith row and jth column as a cross-hair at (i, j). There are mn cross- hairs, one corresponding to each cell of the grid. Each cross-hair has m + n 1 cells. If − there is a k-domatic coloring, then each cross-hair contains all k colors occurs at least once. Claim. In any domatic coloring of G H, each color should appears in at least m cells.

Suppose a color c1 appears in less than m cells, then there exists a row i as well as a column j in the grid which do not contain c1. In this case, the cross-hair at (i, j) does not contain c1 and hence the coloring is not a domatic coloring. Thus the claim holds. Since each color should appears in at least m cells and there are mn cells in the grid, the maximum possible value of k in any domatic k-coloring is n. Thus d(G H) n = ≤ max V (G) , V (H) . {| | | |} Now, let us prove the lower bound for dt(G H). Let r and s be the domatic numbers of

G and H respectively. Suppose r s. Let D ,D ,...,Dr be a domatic partition of V (G). ≥ 1 2 For 1 i r, any vertex u Di and v V (H), let us deﬁne a coloring f for the vertices ≤ ≤ ∈ ∈ of G H by f((u, v)) = i. Since H does not have an isolated vertex, f is a total domatic coloring of G H with r colors and thus dt(G H) r = max d(G),d(H) . ≥ { } If at least one of these graphs G and H is disconnected, then the upper bound can be improved by considering the smallest size of the components of G and H. The bounds given in Theorem 3.1 are tight for the graphs mentioned in Corollary 3.2.

Corollary 3.2. Let m, n be two integers greater than 1 and G be a graph of order m without an isolated vertex. If m n, then dt(G Kn) = d(G Kn) = n. In particular, dt(Km ≤ Kn)= d(Km Kn) = max m, n . { }

Proof. By Theorem 3.1, we have n = max d(G),d(Kn) dt(G Kn) d(G Kn) { } ≤ ≤ ≤ max m, n = n. { }

The tightness of the lower bound for dt(G H) in Theorem 3.1 can be also seen by considering G ∼= Kn and H ∼= K2. More generally, in all cases when G is domatically full and δ(H)=1, the lower bound is attained for dt(G H).

5 Corollary 3.3. If G is a domatically full graph and H is a graph with minimum degree 1, then dt(G H)= d(G).

Proof. We have dt(G H) d(G) by Theorem 3.1. The upper bound follows since dt(G ≥ H) δ(G H)= δ(G)+1= d(G). ≤ The tightnessof the upper bound inTheorem 3.1 can alsobe seen by considering G ∼= Kn and H ∼= K2. Theorem 3.4 demonstrates the same in a more general case.

Theorem 3.4. Let r, s ,s ,...,sr be positive integers such that r 2, s s 0 1 −1 ≥ 0 ≤ 1 ≤···≤ sr−1. If G is a graph with total domatic number at least s0 and H is a graph which contains Ks ,s ,...,s − as a spanning subgraph, then dt(G H) rs . If each si is equal to s and 0 1 r 1 ≥ 0 0 V (G) rs , then dt(G H)= V (H) = rs and dt(H H)= V (H) . If G is a graph | | ≤ 0 | | 0 | | with domatic number at least s0, then the same results hold for d(G H).

Proof. Let G be a graph with total domatic number at least s0 and H be a graph which ′ contains the spanning subgraph H , namely Ks ,s ,...,s − . Let Ui, i Zs be the color classes 0 1 r 1 ∈ 0 corresponding to s0-total domatic coloring of G. We label the vertices in each color class Ui th ′ of G as uij : j Z U and label the vertices of k part of H , k Zr as vkl : l Zs . { ∈ | i|} ∈ { ∈ k } Let us deﬁne a coloring f for the vertices of G H′ in the following way:

f((uij, vkl))=(r(i + l)+ k) (mod rs0).

′ ′ ′ ′ The vertex (uij, vkl) should be adjacent to the vertices (ui j , vkl): i Zs0 , for some j ′ ′ { ′ ∈ ∈ ′ ′ ′ Z|Ui′ | in the layer Gvkl and (uij, vk l ): k Zr,k = k,l Zsk in the layer Huij . Note } ′ ′ { ′ ∈ ′ 6 ∈ } that (r(i + l )+ k ) (mod rs0): k Zr,l Zsk′ = Zrs0 . The set of colors in the { ′ ∈ ∈ } ′ ′ neighbors of (uij, vkl) are (r(i + l)+ k) (mod rs0): i Zs0 Zrs0 (r(i + l )+ k) ′ {{ ∈ } ∪ ′ \{ (mod rs ): l Zs = Zrs as sk s . Thus, each vertex of G H sees all the colors 0 ∈ k }} 0 ≥ 0 ′ Zrs0 in its open neighborhood. Since G H is a spanning subgraph of G H, we have ′ dt(G H) dt(G H ) rs . ≥ ≥ 0 For r 2, i Zr, if each si is equal to s0, then the coloring f mentioned above ≥ ∈ ′ yields that dt(G H) dt(G H ) rs = V (H) and by Theorem 3.1, we have ≥ ≥ 0 | | dt(G H) max V (G) , V (H) = V (H) . Since dt(H) s , by above arguments we ≤ {| | | |} | | ≥ 0 get dt(H H)= V (H) . | | If G is a graph with domatic number at least s0, then the same proof mentioned above will work for d(G H) in such a way that each vertex of G H′ should contain the vertices ′ of all the colors Zrs in its closed neighborhood. Thus d(G H) d(G H ) rs . Also, 0 ≥ ≥ 0 d(G H)= V (H) = rs when each si is equal to s and V (G) rs . | | 0 0 | | ≤ 0

Remark 3.1. A graph H contains Ks ,s ,...,s − , s s sr , as a spanning graph if 0 1 r 1 0 ≤ 1 ≤···≤ −1 and only if the components of H can be grouped into r parts such that each part has at least s0 vertices. Next, let us consider the Cartesian product of complete graphs and cycles. The upper bound given in Theorem 3.1 is not tight for cycles of length larger than the size of the complete graphs.

6 3 b 1 b b 3 1 2 b b b 3 2 2 b b 1 2 b b 3b b 1 2 b b 2 2 b b 2 b b 1 1 b b 3 3

Figure 1: 3-total domatic coloring of G K2, where G is a Peterson graph

Proposition 3.5. Let m, n be two integers such that m > n 3, we have dt(Cm Kn)= n. ≥

Proof. Let U = ui : i Zm and V = vj : j Zn be the vertices of Cm and Kn { ∈ } { ∈ } respectively. Suppose there exists an (n + 1)-total domatic coloring for Cm Kn. Since

Cm Kn is (n + 1)-regular, the neighbors of each vertex should be colored distinctly and any edge of the same color class cannot be in the layer ui Kn for all i, 0 i m 1. Let { } ≤ ≤ − us consider an edge (ui, v )(ui , v ) in Cm v such that both the vertices are colored c . 0 +1 0 { 0} 1 For all x ui ,ui,ui ,ui , the vertices of x Kn, (ui , v ) and (ui , v ) cannot ∈{ −1 +1 +2} { } −2 0 +3 0 be colored with color c1, otherwise there exists a vertex which sees the same color c1 in its two neighbors. If we choose the another edge for the color class c1 either (ui+3,y)(ui+4,y) or (ui ,y)(ui ,y) for some y v ,...,vn , then it covers the vertices of exactly 3 −2 −3 ∈ { 1 −1} new layers z Kn for all z ui ,ui ,ui or z ui ,ui ,ui respectively. { } ∈ { +3 +4 +5} ∈ { −2 −3 −4} Otherwise, an edge will covers 4 new layers of Kn in Cm Kn. Thus an edge which chosen

ﬁrst has covered 4 layers of Kn and the subsequent edges covers either 3 or 4 layers of Kn, m−1 likewise we can choose at most 3 edges for a color class. By (i) of Proposition 2.1, size of an each color class equals mn 2 m−1 2m which in turns n 2(n+1) and n+1 ≤ 3 ≤ 3 ≤ 3 yields n 2, a contradiction. Hence dt(Cm Kn) n and lower bound follows from ≤ ≤ dt(Cm Kn) max d(Cm),d(Kn) = n. ≥ { }

Now, let us consider the bounds for the domatic and total domatic numbers of G K2.

Proposition 3.6. Let G be a graph without an isolated vertex, we have d(G) dt(GK ) ≤ 2 ≤ 2dt(G)+1 2d(G)+1 and d(G K ) 2d(G)+1. ≤ 2 ≤

Proof. Let us consider the graph G K . By Theorem 3.1, dt(G K ) max d(G), 2 = 2 2 ≥ { } d(G). Let k = dt(G K2) and let v1, v2 be the vertices of K2. Let D1,D2,...,Dk be a total domatic partition of V (G K2). It is easy to observe that for any two i, j [k], the set ∈ k u : (u, v) Di Dj is a total dominating set of G. Thus d(G) dt(G) which { ∈ ∪ } ≥ ≥ 2 in turns that k = dt(G K ) 2dt(G)+1 2d(G)+1. Similar arguments hold for the 2 ≤ ≤ domatic partition of V (G K ) and we get d(G K ) 2d(G)+1. 2 2 ≤

There is an example of graph G such that dt(G K2) > d(G). Let G be a Peterson graph, we have dt(G)= d(G)=2 and dt(GK2)=3. The 3-total domatic coloring of GK2 has

7 been shown in Figure 1. By (iii) of Proposition 2.1, 4-total domatic coloring is not possible 2 as 4 ∤ 40, where E(GK ) = 40. The upper bound 2dt(G)+1 for dt(GK ) is tight for | 2 | 2 graphs Kn, n odd and C3k, 4 ∤ k. We could not ﬁnd a graph G such that dt(GK2)=2d(G). This leads us to ask the following question.

Problem 3.7. Let G be a graph without an isolated vertex, ﬁnd the smallest constant c such that dt(G K ) cd(G)+ O(1). 2 ≤ Let G be the complement of a perfect matching. It is easy to observe that d(G) = |V (G)| and d(G K ) = V (G) = 2d(G). Also, we could not ﬁnd a graph G such that 2 2 | | d(G K2)=2d(G)+1. This leads us to ask a question: Is there any graph G such that d(G K2) > 2d(G)? Now, let us consider a lower bound for the domatic and total domatic numbers of Carte- sian product of a graph and a bipartite graph in terms of its domatic and total domatic numbers. The bound given in Theorem 3.8 has been applied multiple times in this paper.

Theorem 3.8. Let G be a graph and H be a bipartite graph, we have

(i) dt(G H) 2 min dt(G),dt(H) and ≥ { } (ii) d(G H) 2 min d(G),dt(H) . ≥ { } Proof. (i) Let G be a graph of order m and H be a bipartite graph of order n with bipartition

[X,Y ]. Let ui : i Zm and vj : j Zn be the vertices of G and H respectively. Let { ∈ } { ∈ } k = min dt(G),dt(H) , there exists a total domatic coloring for G and H with k colors. { } Let g and h be a k-total domatic coloring of G and H respectively, and let Zk be the k colors.

Now, let us deﬁne a coloring f for the vertices of G H. For i Zm and j Zn, ∈ ∈

(2g(ui)+2h(vj)) (mod2k) if vj X f((ui, vj)) = ∈ (3.1) (2g(ui)+2h(vj)+1) (mod2k) if vj Y. ( ∈

For any vertex ui V (G) and vj X, by the coloring f, the colors seen by the vertex ∈ ∈ (ui, vj) from the neighbors of ui in Gv are (2s +2h(vj)) (mod 2k): s Zk = 2l : j { ∈ } { l Zk and from the neighbors of vj in Hu are (2g(ui)+2t +1) (mod 2k): t Zk = ∈ } i { ∈ } 2l + 1 : l Zk . Similarly, for any vertex ui V (G) and vj Y , the colors seen by { ∈ } ∈ ∈ the vertex (ui, vj) from the neighbors of ui in Gv are 2l + 1 : l Zk and from the j { ∈ } neighbors of vj in Hu are 2l : l Zk . Each vertex (ui, vj) in G H sees all the colors i { ∈ } Z2k in its open neighborhood and thus f is a total domatic coloring using 2k colors. Hence dt(G H) 2k = 2 min dt(G),dt(H) . ≥ { } (ii) Let k = min d(G),dt(H) . Let g be a k-domatic coloring of G and h be a k-total { } domatic coloring of H. For any vertex ui V (G) and vj X, by the coloring f deﬁned ∈ ∈ in Equation (3.1), the colors seen by the vertex (ui, vj) from the closed neighborhood of ui in Gv are (2s +2h(vj)) (mod 2k): s Zk = 2l : l Zk and from the open j { ∈ } { ∈ } neighborhood of vj in Hu are (2g(ui)+2t +1) (mod 2k): t Zk = 2l + 1 : l Zk . i { ∈ } { ∈ } Similarly, for any vertex ui V (G) and vj Y , the colors seen by the vertex (ui, vj) from ∈ ∈ 8 the closed neighborhood of ui in Gv are 2l + 1 : l Zk and from the open neighborhood j { ∈ } of vj in Hu are 2l : l Zk . Each vertex (ui, vj) in G H contains vertices of all i { ∈ } the colors Z2k in its closed neighborhood and thus f is a domatic coloring using 2k colors.

Hence d(G H) 2 min d(G),dt(H) . ≥ { } If at least one of these graphs G, H is disconnected, then apply the same technique to each component of G H separately. The bound given in Theorem 3.8 for d(G H) is tight, which follows by taking G ∼= Kn and H ∼= Kn,n and also, the bound for dt(G H) is tight by taking G ∼= K2n and H ∼= Kn,n. One of the simplest examples such that a strict inequality holds in Theorem 3.8 is G ∼= C6n, when n is odd. By Theorem 3.1, we have dt(C6n C6n) d(C6n)=3 but dt(C6n)=1. ≥ Theorem 3.8 is not true for all graphs in general. For G ∼= K2 K3, we have dt(G)=3. Since G G is 6-regular, the neighbors of each vertex should be colored distinctly when dt(G G)=6. There exists a color which occurs at least twice in the subgraph K3 K3 of G G and hence there exists a vertex which contains the same color in its two neighbors, a contradiction. Thus dt(G G) < 6=2dt(G). We can extend the above idea to show that there exists graph G such that dt(G G) d(G G) dt(G)+ O dt(G) . ≤ ≤ p Proposition 3.9. For the graph G = Kn K , dt(GG) d(GG) dt(G)+ 2dt(G). 2 ≤ ≤ p Proof. From Corollary 3.2 we know that dt(G) = d(G) = n and we will show that d(G G) n + √2n. Suppose that there exists a domatic coloring f of G G using more than ≤ n + √2n colors, then there exists a color (call it red) which appears in at most 4(n √2n) − vertices of G G. Otherwise, we get a contradiction, since 4(n √2n +1)(n + √2n +1) = − 4(n2 + 1) > V (G G) . | | Observe that GG is isomorphic to H C4, where H = Kn Kn. We label the vertices of C4 with Z4 and the copy of H corresponding to vertex i of this C4 will be called Hi. Let

Ri denote the set of vertices in Hi which are colored red by f. Since R R | 0 ∪···∪ 3| ≤ 4(n √2n), the smallest of them, without loss of generality say R , has at most n √2n − 0 − vertices. Let k = n R , and note that k √2n. −| 0| ≥ The coloring f on H can be represented by an n n matrix M . Since R = n k, 0 × 0 | 0| − there exists at least k rows and k columns of M0 which do not contain any red vertex. Hence the k2 vertices corresponding to the k k submatrix determined by the above rows and × columns are neither red, nor they see a red neighbor in H0. Hence each of them have a red neighbor in H or H . Since each vertex in V (H ) V (H ) is adjacent to exactly one 1 3 1 ∪ 3 vertex in V (H ), we can conclude that R R k2. But now, since R R , we get 0 | 1 ∪ 3| ≥ | 2|≥| 0| R R 2(n k)+ k2. Since k2 2k is an increasing function of k for k 1, | 0 ∪···∪ 3| ≥ − − ≥ the above lower bound is at least 4n 2√2n for any k √2n. This contradicts our earlier − ≥ upper bound on R R . | 0 ∪···∪ 3| One of the consequences of Theorem 3.8 is Corollary 3.10.

9 Corollary 3.10. Let d be a positive integer, n and k be powers of 2, k n. If G is a graph ≤ with d(G) kd and for 1 i n, Hi is a bipartite graph such that dt(Hi) d, then n ≥ ≤n−k ≤ ≥ dt( Hi) nd and d(G ( Hi)) nd. i=1 ≥ i=1 ≥ n Proof. For each j taken in the increasing order 1, 2,..., 2 , repeated application of The- orem 3.8 to pairs of bipartite graphs that have a total domatic coloring with jd colors yields the bipartite graphs having a total domatic coloring with 2jd colors. Finally, we n get dt( Hi) nd. Now, let G be a graph such that d(G) kd. First we group i=1 ≥ ≥ H1,H2,...,Hn−k into groups of size k each and apply the above argument to get the bi- ′ ′ ′ ′ n partite graphs H ,H ,...,H n with dt(Hj) kd, 1 j 1. Now, consider the 1 2 k −1 k ′ ′ ′ ≥ ≤ ≤ − graphs in this order G,H ,H ,...,H n . Apply Theorem 3.8 (ii) to the ﬁrst pair and Theo- 1 2 k −1 rem 3.8 (i) to the remaining pairs of graphs repeatedly as mentioned above to obtain the ﬁnal n−k n result. Thus d(G ( Hi)) ( )kd = nd. i=1 ≥ k

4 Hamminggraphs

Let us start this section by re-derive the results for the domatic number of the hypercubes

Qn−1 and Qn [31] and the total domatic number of the hypercube Qn [8], when n is a power of 2.

Corollary 4.1. Let n be a powers of 2, we have dt(Qn) = dt(Qn+1) = n and d(Qn−1) = d(Qn)= n.

Proof. It is easy to see that dt(Qn) n and d(Qn−1) n. Since dt(K2)=1, d(K2)=2, n ≤n−2 ≤ Qn = K2 and Qn−1 = K2 ( K2), by Corollary 3.10, we get dt(Qn) n and ∼ i=1 ∼ i=1 ≥ d(Qn−1) n. Suppose dt(Qn+1)= n +1, by (iii) of Proposition 2.1, n +1 should divides n+1 ≥ 2 , a contradiction. Thus dt(Qn ) n. Also, dt(Qn ) dt(Qn) = n. Similarly, by +1 ≤ +1 ≥ Theorem 2.2, we get d(Qn) n and d(Qn) d(Qn )= n. ≤ ≥ −1 27 In [17], Havel obtained that d(Q )=5. Also, 5 = d(Q ) dt(Q ) = 5. We 6 6 ≤ 7 ≤ 24 obtain a new lower bound for d(Qn) and dt(Qn) for some values of n in Corollaryj k 4.2.

k k Corollary 4.2. Let k, n be positive integers such that n 2 7 1, we have d(Qn) 2 5 k ≥ − ≥ and dt(Qn ) 2 5 +1 ≥ k Proof. Let G ∼= Q6 and Hi ∼= Q7, for 1 i 2 . Clearly, d(G)=5 and dt(Hi)=5. By 2k−1 ≤ ≤ 2k k k k Corollary 3.10, we get d(G ( Hi)) 2 5 and dt( Hi) 2 5. Since n 2 7 1, i=1 ≥ i=1 ≥ ≥ − k k we have d(Qn) 2 5 and dt(Qn ) 2 5. ≥ +1 ≥ When n is a power of 2, we can also re-derive the following results on the domination and total domination numbers of the hypercubes Qn−1 [16] and Qn [22] respectively.

10 n γ(Qn) γt(Qn) d(Qn) dt(Qn) 1 1 2 2 1 2 2 2 2 2 3 2 4 4 2 4 4 4 4 4 5 7 8 4 4 6 12 14 5 4 7 16 24 8 5 8 32 32 8 8 9 62 64 8 8 10 107-120 124 8-9 8 11 180-192 214-240 8-11 8-9 12 342-380 360-384 8-11 8-11 13 598-704 684-760 10-13 8-11 14 1171-1408 1196-1408 10-13 10-13 15 211 2342-2816 16 10-13 16 212 212 16 16 17 7377-213 213 16-17 16

Table 1: Bounds on γ(Qn), γt(Qn), d(Qn) and dt(Qn), 1 n 17. Lower bound for ≤ ≤ d(Q13), d(Q14), dt(Q14) and dt(Q15) are new and follow from Corollary 4.2

2k−k−1 2k−k Corollary 4.3. [16, 22] For a positive integer k, γ(Q2k−1)=2 , γt(Q2k )=2 , 2k−k 2k−k+1 γ(Q k ) 2 and γt(Q k ) 2 . 2 ≤ 2 +1 ≤ k− 22 1 2k−k−1 Proof. By Corollary 2.3, 4.1 and (iv) of Proposition 2.1, we get γ(Q2k−1)= 2k =2 k k 22 2k−k 22 and γt(Q2k ) = 2k = 2 respectively. Also, by Corollary 4.1, we have γ(Q2k ) 2k = k ≤ 2k−k 22 +1 2k−k+1 2 and γt(Q k ) k =2 . 2 +1 ≤ 2 2k−k Note that γ(Q2k )=2 follows from the sphere bound mentioned in [28] and γt(Q2k+1)= 2k−k+1 2 follows from the result proved by Azarija et al., [3] namely, γt(Qn+1)=2γ(Qn).

In Table 1, we have mentioned the present best bounds for γ(Qn) (see, Table 1 in [4, 28]) and γt(Qn) follows from the result γt(Qn)=2γ(Qn−1) [3]. The bounds for d(Qn) and dt(Qn) follows from Corollary 4.1 (also see, [8, 31]) and Corollary 4.2. Now, let us start obtain a lower bound for the domatic and total domatic numbers of

Hamming graphs Hn ,q and Hn,q respectively when n is power of 2 and q 2. −1 ≥

Theorem 4.4. Let q be an integer greater than 1 and n is a power of 2, we have d(Hn−1,q) q q ≥ n and dt(Hn,q) n . 2 ≥ 2

Proof. Let K q q be a complete bipartite subgraph of Kq. Any coloring which as- 2 , 2 ⌈ ⌉ ⌊ ⌋q signs same set of different colors to each part of K q q is a total domatic color- 2 2 , 2 q ⌈ ⌉ ⌊ ⌋ ing with 2 colors. Since n is a power of 2, by Corollary 3.10, we have dt(Hn,q) = n n q q dt( Kq) dt( K q , q ) = n . Also, d(Kq) = q 2 , by Corollary 3.10, i=1 ≥ i=1 2 2 2 ≥ 2 ⌈ n⌉−1⌊ ⌋ n−2 q we have d(Hn−1,q)= d( Kq) d(Kq ( K q , q )) n 2 . i=1 ≥ i=1 ⌈ 2 ⌉ ⌊ 2 ⌋ ≥ 11 Note that an equality holds for the graphs H1,q, H2,2q, and the problem remains open for

Hn,q, n 4 except Hn, . ≥ 2 In [8] Chen et al., obtained the injective chromatic number of Hn,q as follows.

(qk−1) Theorem 4.5. [8] Let k, n be positive integers. If q is a prime power and n = (q−1) , then 2 k χi(Hn,q)= χ(Hn,q)= q .

Corollary 4.6 is an immediate consequence of Theorem 3.1 and Theorem 4.5.

(qk−1) Corollary 4.6. Let k, n be positive integers. If q is a prime power and n = (q−1) , then k k d(Hn,q)= q and dt(Hn ,q) q . +1 ≥ 2 k Proof. By Theorem 4.5, we have χi(Hn,q) = χ(H ) = q = n(q 1)+1. Since Hn,q n,q − is n(q 1)-regular and there is a proper injective coloring with n(q 1)+1 colors, by − k − Corollary 2.3 we get d(Hn,q)= q . By Theorem 3.1, we have dt(Hn+1,q)= dt(Hn,q Kq) k ≥ d(Hn,q)= q .

As a consequence of Theorem 3.1, 4.4 and Corollary 3.10, 4.6, we get Corollary 4.7

Corollary 4.7. Let n be a positive integer and q be a prime power. If k is the largest positive k integer such that q −1 n, j is the smallest positive integer such that qk 2j q and i q−1 ≤ ≤ 2 qk−1 i j i k is the largest positive integer such that q−1 + (2 1)2 n, then d(Hn,q) 2q and i k − ≤ ≥ dt(Hn ,q) 2 q . +1 ≥ ′ qk−1 i Proof. Let n = and G = Hn′,q. For 0 l < 2 , let Gl = H j . It is clear from q−1 ∼ ≤ ∼ 2 ,q the proof of Theorem 4.4, there exists a spanning bipartite subgraph of Gl that have a total j q j q k domatic coloring with 2 2 colors, let it be Hl. Clearly, dt(Hl) 2 2 q and by 2i−≥1 ≥ k i k Corollary 4.6, d(G) = q. By Corollary 3.10, we get d(G ( Hl)) 2 q . Since l=1 ≥ 2i−1 2i−1 ′ i j i k n n + (2 1)2 , we have d(Hn,q) d(Hn′,q ( Gl)) d(G ( Hl)) 2 q . ≥ − ≥ l=1 ≥ l=1 ≥ i k By Theorem 3.1, we have dt(Hn ,q) d(Hn,q) 2 q . +1 ≥ ≥

5 Tori

In this section, we examine the d-dimensional tori which are domatically and total domatically full. First we obtain some sufﬁcient conditions for the tori which are total domatically full.

Corollary 5.1 as an immediate consequence of Corollary 3.10 and the fact that dt(Cn)=2 if and only if n 0 (mod 4). ≡

Corollary 5.1. Let d,k1,k2,...,kd be positive integers such that d isa power of 2 and ki 0 d d d ≡ (mod 4), 1 i d, we have dt( Cki )=2d and γt( Cki )=( ki)/2d. ≤ ≤ i=1 i=1 i=1 Q

12 In the remaining part of this section, we try to generalize this result to larger collections of tori. In [13], Gravier independently obtained the total domination number of some tori by the concept of periodic tiling which mentioned in [14].

Theorem 5.2. [13] Let d,k1,k2,...,kd be positive integers such that d 2 and ki 0 d d ≥ ≡ (mod 4d), for 1 i d, we have γt( Cki )=( ki)/2d. Moreover, if d is even and for ≤ ≤ i=1 i=1 any positive integer ki, 1 i d such that ki 0Q (mod 2d), then this equality still holds. ≤ ≤ ≡ By vertex transitivity of the tori, one can obtain a total domatic coloring of the tori mentioned in Theorem 5.2 with 2d colors. Hence this tori are total domatically full. We extend this to larger classes of tori. Moreover, our proof is much shorter.

Theorem 5.3. Let d,k1,k2,...,kd be positive integers. If kd is congruent to 0 (mod 4) and d the remaining ki’s are congruent to 0 (mod 2d), then dt( Ck )=2d. i=1 i

d Proof. Let G = Ck , where d, ki’s are positive integers. Since δ(G)=2d, it is enough to ∼ i=1 i give a 2d-total domatic coloring for G. Let the vertices of G be x =(x , x ,...,xd): xi { 1 2 ∈ Zk , 1 i d . Now, let us define a coloring f by i ≤ ≤ } d−1 i=1 i xi (mod 2d) if xd 0or1 (mod 4) f(x)= d−1 ≡ ( i xi)+ d (mod 2d) if xd 2or3 (mod 4). ( P i=1 ≡ Let k be the color of theP vertex x =(x1, x2,...,xd) defined by f. The set of neighbors of x are (x ,...,xi , xi 1, xi ,...,xd), (x ,...,xi , xi +1, xi ,...,xd) :1 i d . {{ 1 −1 − +1 1 −1 +1 } ≤ ≤ } The set of colors in the neighbors of x are the union of k i, k + i along the dimension { − } i, 1 i d 1 and k,k + d along the dimension d which equals Z d. Thus each vertex ≤ ≤ − { } 2 contains vertices of all 2d colors Z2d in its open neighborhood and f is a 2d-total domatic d coloring for G. Hence, dt( Cki )=2d. i=1 In Theorem 5.3, d-dimensional tori are total domatically full when a cycle of length is a multiple of 4 and other cycles length are a multiple of 2d but this condition can be further generalized. We obtain a sufficient condition for the total domatically full tori in Corollary 5.4 which generalize the results mentioned in Corollary 5.1 and Theorem 5.3.

p p Corollary 5.4. Let d =2 q, q be an odd integer and p 0. If 2 number of ki’s are congru- ≥ d ent to 0 (mod 4) and the remaining ki’s are congruent to 0 (mod 2q), then dt( Ck )=2d. i=1 i

d Proof. Let G = Ck , where d, ki’s are positive integers. For p = 0, the proof follows ∼ i=1 i d from Theorem 5.3. Let us consider p 1. Since Ck is transitive with respect to the ≥ i=1 i p product, among 2 number of ki’s are congruent to 0 (mod 4), without loss of generality, d p choose those ki’s are kq,k2q,...,kd. Now, split the graph Ck into product of 2 smaller i=1 i

13 q 2q d 2p product of q graphs each as ( Cki ) ( Cki ) ( Cki ) = Gj, where i=1 i=q+1 ··· i=d−q+1 ∼ j=1 p d = 2 q. By Theorem 5.3, each smaller product graph Gj has the total domatic number 2q d 2p p p for 1 j 2 . By Corollary 3.10, we have dt( Cki )= dt( Gj)=2 2q =2d. ≤ ≤ i=1 j=1 The Corollary 5.5 is an immediate consequence of Corollary 5.4 and (iv) of Proposition 2.1 which turns to be a generalization of the result given in Corollary 5.1 and a result due to S. Gravier [13] for the d-dimensional tori given in Theorem 5.2.

p p Corollary 5.5. Let d =2 q, q be an odd integer and p 0. If 2 number of ki’s are congru- ≥ d ent to 0 (mod 4) and the remaining ki’s are congruent to 0 (mod 2q), then γt( Cki ) = i=1 d ( ki)/2d. i=1 Q Note that, the family of tori given in Corollary 5.5 contains the family of tori as mentioned in Theorem 5.2. Suppose the dimension of the torus is 12, Corollary 5.5 ﬁnd γt for the tori 12 Cpi , where four pi’s are congruent to 0 (mod 4) and the remaining pi’s are congruent to i=1 12 0 (mod 6) but Theorem 5.2 ﬁnds γt for the torus Cki , where each ki 0 (mod 24). i=1 ≡ Corollary 5.6 is an immediate consequence of Corollary 5.4, 5.5.

p Corollary 5.6. Let d = 2 q, q be an odd integer and p 0 and let Gi be a Hamiltonian p ≥ graph, 1 i d. If 2 number of Gi’s have V (Gi) 0 (mod 4) and the remaining Gi’s ≤ ≤ d | | ≡ d d have V (Gi) 0 (mod 2q), then dt( Gi) 2d and γt( Gi) ( V (Gi) )/2d. | | ≡ i=1 ≥ i=1 ≤ i=1 | | Q Next, we obtain a sufﬁcient condition for the domatically full tori.

Theorem 5.7. Let d,k1,k2,...,kd be positive integers. If each ki is congruent to 0 (mod 2d+ d 1), then d( Ck )=2d +1. i=1 i

d Proof. Let G = Ck , where d, ki’s are positive integers and each ki 0 (mod 2d + 1). ∼ i=1 i ≡ Since δ(G)+1=2d +1, it is enough to give a (2d + 1)-domatic coloring for G. Let the vertices of G be x =(x , x ,...,xd): xi Zk , 1 i d . Now, let us deﬁne a coloring { 1 2 ∈ i ≤ ≤ } f by

d f(x)= i=1 i xi (mod 2d + 1).

Let k be the color of the vertex x =(Px1, x2,...,xd) deﬁned by f. The set of neighbors of x are (x ,...,xi , xi 1, xi ,...,xd), (x ,...,xi , xi +1, xi ,...,xd) :1 i d . {{ 1 −1 − +1 1 −1 +1 } ≤ ≤ } The set of colors in the neighbors of x are k i, k + i along the dimension i, 1 i d { − } ≤ ≤ which equals Z2d+1 k . Thus each vertex contains vertices of all 2d +1 colors Z2d+1 in \{ } d its closed neighborhood. Hence f is a (2d + 1)-domatic coloring for G and d( Cki ) = i=1 2d +1.

14 As a consequence of Theorem 5.7 and Corollary 2.3, we get Corollary 5.8 which re- derive the result proved by Klavžar and Seifter [23].

Corollary 5.8. [23] Let d,k1,k2,...,kd be positive integers. If each ki is congruent to 0 d d (mod 2d + 1), then γ( Cki )=( ki)/(2d + 1). i=1 i=1 Q Corollary 5.9 is an immediate consequence of Theorem 5.7 and Corollary 5.8.

Corollary 5.9. Let d be a positive integer and let Gi be a Hamiltonian graph, 1 i d. d ≤d ≤ If each Gi have V (Gi) 0 (mod 2d + 1), then d( Gi) 2d +1 and γ( Gi) | | ≡ i=1 ≥ i=1 ≤ d ( V (Gi) )/(2d + 1). i=1 | | Q 6 Conclusion and open problems

Accepting the invitation by Chen, Kim, Tait and Verstraete [8] to determine any relationships between dt(G G) and dt(G), we started this investigation aiming to ﬁnd good lower and upper bounds to dt(G G) in terms of dt(G). In this paper, we have made improvements to the easy lower bound dt(G G) dt(G). Firstly we showed that if δ(G) 1, then ≥ ≥ dt(GG) d(G). We also showed that dt(GG) 2dt(G) if G is bipartite. Bipartiteness ≥ ≥ is necessary for the above lower bound. We can show the existence of inﬁnite families of non- bipartite graphs where dt(G G) = dt(G)+ 2dt(G). Nevertheless, we have indirectly ′ ′ used this result for non-bipartite graphs in the form dt(G G) 2dt(G ), where G is a p ≥ spanning bipartite subgraph of G.

In contrast, we haven’t been able to prove any upper bound for dt(G G) in terms of dt(G). We know of graphs G (G = K2n+1 for example) where dt(G G)=2dt(G)+1. We conjecture that it is the maximum possible.

Conjecture 6.1. For any two graphs G and H without an isolated vertex,

dt(G H) 2 max dt(G),dt(H) +1. ≤ { }

A weaker form of the above conjecture, dt(G H) 2 max d(G),d(H) +1, is also ≤ { } an interesting open problem. As far as we know, the version of Conjecture 6.1 for domatic number is also open.

Conjecture 6.2. For any two graphs G and H without an isolated vertex,

d(G H) 2 max d(G),d(H) +1. ≤ { }

If this upper bound is true, then it is also tight. To see that, consider the cycle Cn, where n is a multiple of 5 but not 3. By Theorem 5.7, we have d(Cn Cn)=5=2d(Cn)+1.

15 Acknowledgment

First author’s research was supported by Post Doctoral Fellowship at Indian Institute of Technology, Palakkad. References

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