Properties of the Roman (k, k)-domatic number

In this section we present basic properties of d^k_R(G) and sharp bounds on the Roman(k, k)-domatic number of a graph.

Theorem 2.1. Let G be a graph of order n with Roman k-domination number γkR(G)and Roman (k, k)-domatic number d^k_R(G). Then

The Roman(k, k)-domatic number of a graph 51 Proof. LetγkR(G) =nandd^k_R(G) = 2k. It follows from Proposition B thatGdoes not contain a bipartite subgraphH with bipartitionX, Y such that|X|>|Y| ≥k anddeg_H(v)≥kfor eachv∈X. Let{f1, . . . , f2k} be a Roman(k, k)-dominating family on G. By Theorem 2.1,γ_kR(G) =ω(f_i) =n for each i. First suppose for eachi, there exists a vertexxsuch thatf_i(x)6= 1. Assume thatH_i is a subgraph of G with vertex set V₀^fi∪V₂^fi and edge set E(V₀^fi, V₂^fi). Since ω(fi) = n and fi is a Roman k-dominating function, |V₂^fi| = |V₀^fi| and deg_Hi(v) ≥ k for each v ∈V₀^fi. By Theorem 2.1,P2k

i=1f_i(v) = 2k for eachv∈V(G)which implies that

|{i| v ∈V₂^fi}|=|{i| v ∈V₀^fi}|=k for each v ∈V(G). Now supposefi(x) = 1 for each x∈V(G) and somei, say i= 2k. Define the bipartite subgraphsHi for 1≤i≤2k−1as above.

Conversely, assume that Gdoes not contain a bipartite subgraph H with bi-partition X, Y such that |X|>|Y| ≥ k and deg_H(v) ≥k for each v ∈X and G has2k or2k−1connected bipartite subgraphsHi = (Xi, Yi)with|Xi|=|Yi|and deg_H

i(v)≥k for eachv ∈X_i. Then by Proposition B,γ_kR(G) =n. If Ghas2k connected bipartite subgraphsH_i, then the mappingsf_i:V(G)→ {0,1,2}defined by

fi(u) = 2 ifu∈Yi, fi(v) = 0 if v∈Xi,andfi(x) = 1 for eachx∈V −(Xi∪Yi) are Romank-dominating functions onGand{fi |1≤i≤2k} is a Roman(k, k)-dominating family on G. If Ghas2k−1 connected bipartite subgraphsHi, then the mappingsfi, g:V(G)→ {0,1,2} defined byg(x) = 1 for eachx∈V(G)and

f_i(u) = 2 ifu∈Y_i, f_i(v) = 0 if v∈X_i,andf_i(x) = 1 for eachx∈V −(X_i∪Y_i) are Roman k-dominating functions onGand {g, fi |1≤i≤2k−1} is a Roman (k, k)-dominating family onG.

Thusd^k_R(G)≥2k. It follows from Theorem 2.1 thatd^k_R(G) = 2k, and the proof is complete.

The next corollary is an immediate consequence of Proposition C, Observation 1.3 and Theorem 2.1.

Corollary 2.3. For every graphGof ordern,d^k_R(G)≤max{∆, k−1}+k.

LetA1∪A2∪. . .∪Ad be ak-domatic partition ofV(G)intok-dominating sets such that d=dk(G). Then the set of functions {f1, f2, . . . , fd} with fi(v) = 2if v∈Aiandfi(v) = 0otherwise for1≤i≤dis an R(k, k)D family onG. This shows thatdk(G)≤d^k_R(G)for every graphG. SinceγkR(G)≥min{n, γk(G)+k}(cf. [6]), for each graphGof ordern≥2, Theorem 2.1 implies thatd^k_R(G)≤ _min{n,γ^2kn

k(G)+k}. Combining these two observations, we obtain the following result.

Corollary 2.4. For any graphGof ordern, d_k(G)≤d^k_R(G)≤ 2kn

min{n, γk(G) +k}.

Theorem 2.5. Let Kn be the complete graph of ordernand ka positive integer.

Thend^k_R(Kn) =nifn≥2k,d^k_R(Kn)≤2k−1 ifn≤2k−1 andd^k_R(Kn) = 2k−1 if k≥2 and2k−2≤n≤2k−1.

Proof. By Proposition F, we may assume that k ≥ 2. Assume that V(K_n) = {x1, x₂, ..., x_n}. First let n≥2k. Since Observation 1.9 implies that γ_kR(K_n) = 2k, it follows from Theorem 2.1 that d^k_R(K_n) ≤ n. For 1 ≤ i ≤ n, define now f_i:V(K_n)→ {0,1,2}by

fi(xi) =fi(xi+1) =. . .=fi(x_i+k−1) = 2 andfi(x) = 0 otherwise,

where the indices are taken modulo n. It is easy to see that {f1, f2, . . . , fn} is an R(k, k)D family onGand henced^k_R(Kn)≥n. Thusd^k_R(Kn) =n.

Now letn≤2k−1. Then Observation 1.9 yields γkR(Kn) =n, and it follows from Theorem 2.1 that d^k_R(Kn) ≤ 2k. Suppose to the contrary that d^k_R(Kn) = 2k. Then by Theorem 2.1, each Roman k-dominating functionfi in any R(k, k)D family {f1, f2, , . . . , f2k} on G is aγkR(G)-function. This implies that fi(x) = 1 for each x ∈V(K_n). Hencef₁ ≡f₂ ≡ · · · ≡ f_2k which is a contradiction. Thus d^k_R(K_n)≤2k−1.

In the special casek≥2and2k−2≤n≤2k−1, Observation 1.4 shows that d^k_R(K_n)≥2k−1 and sod^k_R(K_n) = 2k−1.

In view of Proposition G and Theorem 2.1 we obtain the next upper bounds for the Roman (k, k)-domatic number of complete bipartite graphs.

Corollary 2.6. Let Kp,q be the complete bipartite graph of order p+q such that q ≥ p ≥ 1, and let k be a positive integer. Then d^k_R(Kp,q) ≤ 2k if p < k or q=p=k,d^k_R(Kp,q)≤ ^2k(p+q)_k+p ifp+q≥2k+1andk≤p≤3kandd^k_R(Kp,q)≤ ^p+q₂ if p≥3k.

For some special cases of complete bipartite graphs, we can prove more.

Corollary 2.7. Let K_p,p be the complete bipartite graph of order2p, and letk be a positive integer. Ifp≥3k, then d^k_R(K_p,p) =p. Ifp < k, thend^k_R(K_p,p)≤2k−1.

In particular, if p = k−1, then d^k_R(K_p,p) = 2k−1, and if p = k−2, then d^k_R(K_p,p) = 2k−2.

Proof. Assume first thatp≥3k. LetX ={u1, u2, . . . , up}andY ={v1, v2, . . . , vp} be the partite sets of the complete bipartite graph Kp,p. For 1 ≤ i ≤ p, define fi:V(Kp,p)→ {0,1,2} by

fi(ui) =fi(ui+1) =. . .=fi(u_i+k−1) =fi(vi) =fi(vi+1) =. . .=fi(v_i+k−1) = 2 and fi(x) = 0 otherwise, where the indices are taken modulo p. It is a simple matter to verify that {f1, f2, . . . , fp} is an R(k, k)D family on Kp,p and hence d^k_R(K_p,p)≥p. Using Corollary 2.6 forp=q≥3k, we obtaind^k_R(K_p,p) =p.

Assume next that p < k. Since k > p= ∆(K_p,p), it follows from Observation 1.3 that d^k_R(K_p,p)≤2k−1.

The Roman(k, k)-domatic number of a graph 53 Assume now thatp=k−1. Thenk≥2andn(Kp,p) = 2k−2, and we deduce from Observation 1.4 that d^k_R(Kp,p)≥2k−1and so d^k_R(Kp,p) = 2k−1.

Finally, assume that p = k−2. Then k ≥ 3 and n(K_p,p) = 2k −4. It follows from Observation 1.6 that d^k_R(K_p,p) ≥ 2k−2 and from Observation 1.7 that d^k_R(K_p,p)≤2k−2 and thusd^k_R(K_p,p) = 2k−2.

Theorem 2.8. If Gis a graph of order n≥2, then

γkR(G) +d^k_R(G)≤n+ 2k (2.1) with equality if and only if γ_kR(G) = n and d^k_R(G) = 2k or γ_kR(G) = 2k and d^k_R(G) =n.

Proof. Ifd^k_R(G)≤2k−1, then obviouslyγkR(G) +d^k_R(G)≤n+ 2k−1. Let now d^k_R(G)≥2k. IfγkR(G)≥2k, Theorem 2.1 implies thatd^k_R(G)≤n. According to Theorem 2.1, we obtain

γ_kR(G) +d^k_R(G)≤ 2kn

d^k_R(G)+d^k_R(G). (2.2) Using the fact that the functiong(x) =x+(2kn)/xis decreasing for2k≤x≤√

2kn and increasing for √

2kn ≤ x ≤ n, this inequality leads to the desired bound immediately.

Now let γkR(G) ≤ 2k−1. Since min{n, γk(G) +k} ≤ γkR(G), we deduce that γkR(G) = n. According to Theorem 2.1, we obtain d^k_R(G) ≤ 2k and hence d^k_R(G) = 2k. Thus

γ_kR(G) +d^k_R(G) =n+ 2k.

IfγkR(G) =nandd^k_R(G) = 2korγkR(G) = 2kandd^k_R(G) =n, then obviously γkR(G) +d^k_R(G) =n+ 2k.

Conversely, let equality hold in (2.1). It follows from (2.2) that n+ 2k=γkR(G) +d^k_R(G)≤ 2kn

d^k_R(G)+d^k_R(G)≤n+ 2k, which implies thatγ_kR(G) = _d^2kn_k

R(G) andd^k_R(G) = 2kord^k_R(G) =n. This completes the proof.

The special casek= 1of the next result can be found in [8].

Theorem 2.9. For every graph Gand positive integerk, d^k_R(G)≤δ(G) + 2k.

Moreover, the upper bound is sharp.

Proof. If d^k_R(G)≤2k, the result is immediate. Let now d^k_R(G)≥2k+ 1 and let {f1, f2, . . . , fd}be an R(k, k)D family onGsuch thatd=d^k_R(G). Assume thatvis a vertex of minimum degreeδ(G). Let`be the number of sumsP

u∈N[v]fi(u) = 1 and let m be the number of those sums in which P

u∈N[v]f_i(u) = 2. Obviously, l+ 2m≤2k.

We may assume, without loss of generality, that the equalityP

u∈N[v]f_i(u) = 1

Ifm= 0, then the above inequality chain leads to d≤δ(G) + 1 +`−`/(2k).

Since the last fraction in the sum is a rational number in [0,1]and since m≥1, we deduce that

d≤δ(G) + (`+m) +2k−`−2m

2k ≤δ(G) + (`+m) + 1≤δ(G) +`+ 2m≤δ(G) + 2k as desired.

To prove the sharpness of this inequality, letGi be a copy ofK_k3+(2k+1)k with vertex set V(Gi) ={vⁱ₁, v₂ⁱ, . . . , v_kⁱ3+(2k+1)k} for1 ≤i≤k and let the graphGbe obtained from ∪^k_i=1G_i by adding a new vertex v and joining v to each v₁ⁱ, . . . , vⁱ_k.

The Roman(k, k)-domatic number of a graph 55 Define the Roman k-dominating functionsf_i^s, hl for1≤i≤k,0≤s≤k−1 and 1≤l≤2kas follows:

f_i^s(vⁱ₁) =· · ·=f_i^s(vⁱ_k) = 2, f_i^s(v^j_(i−1)k2+(s+1)k+1) =· · ·=f_i^s(v_(i−1)k^j 2+(s+1)k+k) = 2 ifj∈ {1,2, . . . , k} − {i}andf_i^s(x) = 0 otherwise

and for1≤l≤2k,

hl(v) = 1, hl(vⁱ_k3+lk+1) =. . .=hl(v_kⁱ3+lk+k) = 2 (1≤i≤k), andh_l(x) = 0 otherwise.

It is easy to see thatf_i^sandg_lare Romank-dominating function onGfor each 1≤i≤k,0≤s≤k−1,1≤l≤2k and{f_i^s, g_l|1≤i≤k,0≤s≤k−1 and 1≤ l ≤ 2k} is a Roman (k, k)-dominating family on G. Since δ(G) = k², we have d^k_R(G) =δ(G) + 2k.

For regular graphs the following improvement of Theorem 2.9 is valid.

Theorem 2.10. Let kbe a positive integer. If Gis aδ(G)-regular graph, then d^k_R(G)≤max{2k−1, δ(G) +k} ≤δ(G) + 2k−1.

Proof. If k > ∆(G) = δ(G) then by Observation 1.7, d^k_R(G) ≤ 2k−1 and the desired bound is proved. Ifk≤∆(G), then it follows from Corollary 2.3 that

d^k_R(G)≤δ(G) +k, and the proof is complete.

As an application of Theorems 2.9 and 2.10, we will prove the following Nord-haus-Gaddum type result.

Theorem 2.11. Let k≥1 be an integer. IfGis a graph of ordern, then

d^k_R(G) +d^k_R(G)≤n+ 4k−2, (2.3) with equality only for graphs with ∆(G)−δ(G) = 1.

Proof. It follows from Theorem 2.9 that

d^k_R(G) +d^k_R(G)≤(δ(G) + 2k) + (δ(G) + 2k) = (δ(G) + 2k) + (n−∆(G)−1 + 2k).

If Gis not regular, then ∆(G)−δ(G)≥1, and hence this inequality implies the desired boundd^k_R(G) +d^k_R(G)≤n+ 4k−2. IfGis δ(G)-regular, then we deduce from Theorem 2.10 that

d^k_R(G) +d^k_R(G)≤(δ(G) + 2k−1) + (δ(G) + 2k−1) =n+ 4k−3, and the proof of the Nordhaus-Gaddum bound (2.3) is complete. Furthermore, the proof shows that we have equality in (2.3) only when∆(G)−δ(G) = 1.

Corollary 2.12 ([8]). For every graphGof ordern, dR(G) +dR(G)≤n+ 2, with equality only for graphs with ∆(G) =δ(G) + 1.

For regular graphs we prove the following Nordhaus-Gaddum inequality.

Theorem 2.13. Letk≥1be an integer. IfGis aδ-regular graph of ordern, then d^k_R(G) +d^k_R(G)≤max{4k−2, n+ 2k−1, n+ 3k−2−δ,3k+δ−1}. (2.4) Proof. Letδ(G) =δ andδ(G) =δ. We distinguish four cases.

Ifk≥δ+ 1 andk≥δ+ 1, then it follows from Observation 1.7 that d^k_R(G) +d^k_R(G)≤(2k−1) + (2k−1) = 4k−2.

Ifk≤δandk≤δ, then Corollary 2.3 implies that

d^k_R(G) +d^k_R(G)≤(δ+k) + (δ+k) =δ+ 2k+n−1−δ=n+ 2k−1.

Ifk≥δ+ 1andk≤δ, then we deduce from Observation 1.7 and Corollary 2.3 that

d^k_R(G) +d^k_R(G)≤(2k−1) + (δ+k) = 3k−1 +n−1−δ=n+ 3k−2−δ.

Ifk≤δandk≥δ+ 1, then Observation 1.7 and Corollary 2.3 lead to d^k_R(G) +d^k_R(G)≤(δ+k) + (2k−1) = 3k+δ−1.

This completes the proof.

IfGis aδ-regular graph of ordern≥2, then Theorem 2.13 leads to the following improvement of Theorem 2.11 for k≥2.

Corollary 2.14. Let k≥2be an integer. IfGis aδ-regular graph of ordern≥2, then

d^k_R(G) +d^k_R(G)≤n+ 4k−4.

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