Signed total (j,k)-domatic numbers of graphs

Signed total (j, k)-domatic numbers of graphs

Lutz Volkmann

Lehrstuhl II f¨ur Mathematik RWTH Aachen University

52056 Aachen Germany

volkm@math2.rwth-aachen.de

Abstract

LetG be a finite and simple graph with vertex set V(G), and let f : V(G) → {−1,1} be a two-valued function. If k ≥ 1 is an integer and

x∈N(v)f(x) ≥ k for each v ∈ V(G), where N(v) is the neigh- borhood of v, then f is a signed total k-dominating function onG. A set {f₁, f₂, . . . , f_d} of distinct signed totalk-dominating functions onG with the property that _d

i=1f_i(x) ≤ j for each x ∈V(G), is called a signed total (j, k)-dominating family (of functions) onG, where j≥1 is an integer. The maximum number of functions in a signed total (j, k)- dominating family onGis the signed total (j, k)-domatic number of G, denoted byd^(j,k)_st (G).

In this paper we initiate the study of the signed total (j, k)-domatic number. We present different bounds on d^(j,k)_st (G), and we determine the signed total (j, k)-domatic number for special graphs. Some of our results are extensions of well-known properties of different other signed total domatic numbers.

1 Terminology and introduction

Various numerical invariants of graphs concerning domination were introduced by means of dominating functions and their variants (see, for example, Haynes, Hedet- niemi and Slater [1, 2]). In this paper we define thesigned total(j, k)-domatic number in an analogous way as Henning [4] has introduced the signed total domatic number.

We consider finite, undirected and simple graphs Gwith vertex set V(G) =V and edge setE(G) = E. The cardinality of the vertex set of a graph Gis called the order of G and is denoted by n(G) = n. If v ∈ V(G), then N_G(v) = N(v) is the open neighborhoodof v, i.e., the set of all vertices adjacent to v. The closed neighborhoodN_G[v] =N[v] of a vertexvconsists of the vertex setN(v)∪ {v}. The number d_G(v) = d(v) = |N(v)| is the degree of the vertex v. The minimumand maximum degreeof a graphGare denoted byδ(G) and Δ(G). Thecomplementof a graphGis denoted byG. Afanis a graph obtained from a path by adding a new

vertex and edges joining it to all the vertices of the path. IfA⊆V(G) andf is a mapping fromV(G) into some set of numbers, thenf(A) =

x∈Af(x).

Ifk ≥1 is an integer, then thesigned totalk-dominating function(STkD func- tion) is defined in [7] as a two-valued function f : V(G) → {−1,1} such that

x∈N(v)f(x)≥kfor eachv∈V(G). The sumf(V(G)) is called the weightw(f) of f. The minimum of weightsw(f), taken over all signed totalk-dominating functions f on G, is called the signed total k-domination number of G, denoted by γ_st^k(G).

Aγ_st^k(G) function is a STkD-function on G of weight γ_st^k(G). As the assumption δ(G) ≥ k is necessary, we always assume that when we discuss γ_st^k(G), all graphs involved satisfyδ(G)≥kand thusn(G)≥k+ 1. The function assigning +1 to every vertex ofGis a STkD function, called the function, of weightn. Thusγ_st^k(G)≤n for every graph of order n with δ(G) ≥ k. Moreover, the weight of every STkD function different fromis at most n−2 and more generally, γ_st^k(G)≡n (mod 2).

Henceγ_st^k(G) =nif and only ifis the unique STkD function ofG. The special case k= 1 was defined by Zelinka in [8], and has been studied by several authors (see for example, Henning [3]). We make use of the following result.

Observation 1. ([6])LetGbe a graph of ordernand minimum degreeδ(G)≥k.

Thenγ_st^k(G) =n if and only if for eachv ∈V(G), there exists a vertexu∈N(v) such thatk≤d(u)≤k+ 1.

Letj≥1 be an integer. A set{f₁, f₂, . . . , f_d}of distinct signed totalk-dominating functions onGwith the property thatd

i=1f_i(x)≤jfor eachx∈V(G), is called a signed total(j, k)-dominating family onG. The maximum number of functions in a signed total (j, k)-dominating family on Gis thesigned total(j, k)-domatic number of G, denoted byd^(j,k)_st (G). The signed total (j, k)-domatic number is well-defined andd^(j,k)_st (G) ≥ 1 for all graphs G, since the set consisting of any STkD function, for instance the function , forms a signed total (j, k)-dominating family of G. A d^(j,k)_st (G)-family of a graph G is a signed total (j, k)-dominating family containing d^(j,k)_st (G) STkD functions. The special casesj = k = 1 or j = 1 or j = k of this paprameter are investigated by Henning [4] or Khodkar and Sheikholeslami [5] or Sheikholeslami and Volkmann [6].

Observation 2. LetGbe a graph of ordernandδ(G) ≥k. If γ_st^k(G) =n, then is the unique STkD function ofGand sod^(j,k)_st (G) = 1.

The following observation is a consequence of Observations 1 and 2.

Observation 3. If Gis a graph of ordern≥3 andk =n−1 ork =n−2, then γ_st^k(G) =nand thusd^(j,k)_st (G) = 1.

2 Properties of the signed total (j, k)-domatic number

In this section we present basic properties of d^(j,k)_st (G) and bounds on the signed total (j, k)-domatic number of a graph. Some of our results are extensions of these

given by Henning [4], Khodkar and Sheikholeslami [5] as well as Sheikholeslami and Volkmann [6].

Theorem 4. IfGis a graph of ordernwith minimum degreeδ(G)≥k, then γ_st^k(G)·d^(j,k)_st (G)≤j·n.

Moreover, ifγ_st^k(G)·d^(j,k)_st (G) =j·n, then for eachd^(j,k)_st (G)-family {f₁, f₂, . . . , f_d} withd=d^(j,k)_st (G) onG, each functionf_i is aγ_st^k(G)-function and_d

i=1f_i(x) =jfor allx∈V(G).

Proof. If {f₁, f₂, . . . , f_d} is a signed total (j, k)-dominating family onG such that d=d^(j,k)_st (G), then the definitions imply

d·γ_st^k(G) =

d

i=1

γ_st^k(G)≤

d

i=1

x∈V(G)

f_i(x)

=

x∈V(G)

d

i=1

f_i(x)≤

x∈V(G)

j=j·n.

If γ_st^k(G)·d^(j,k)_st (G) = j ·n, then the two inequalities occurring in the proof be- come equalities. Hence for thed^(j,k)_st (G)-family{f₁, f₂, . . . , f_d}onGand for eachi,

x∈V(G)f_i(x) = γ^k_st(G), and thus each function f_i is a γ_st^k(G)-function and

_d

i=1f_i(x) =jfor allx∈V(G).

Theorem 5. IfGis a graph with minimum degreeδ(G)≥k, then d^(j,k)_st (G)≤ j·δ(G)

k .

Moreover, ifd^(j,k)_st (G) =j·δ(G)/k, then for each function of any signed total (j, k)- dominating family{f₁, f₂, . . . , f_d}withd=d^(j,k)_st (G), and for all verticesvof degree δ(G),

x∈N(v)f_i(x) =kand_d

if_i(x) =jfor everyx∈N(v).

Proof. Let{f₁, f₂, . . . , f_d}be a signed total (j, k)-dominating family onGsuch that d=d^(j,k)_st (G). Ifv∈V(G) is a vertex of minimum degreeδ(G), then it follows that

d·k =

d

i=1

k≤

d

i=1

x∈N(v)

f_i(x)

=

x∈N(v)

d

i=1

f_i(x)≤

x∈N(v)

j=jδ(G),

and this implies the desired upper bound on the signed total (j, k)-domatic number.

Ifd^(j,k)_st (G) =j·δ(G)/k, then the two inequalities occuring in the proof become equalities, which leads to the two properties given in the statement.

The special casesj=k= 1 orj= 1 orj=kof Theorems 4 and 5 can be found in [4] or [5] or [6]. The upper bound on the productγ_st^k(G)·d^(j,k)_st (G) leads to a bound on the sum of these terms.

Corollary 6.IfGis a graph of ordernwith minimum degreeδ(G)≥k, then γ_st^k(G) +d^(j,k)_st (G)≤jn+ 1.

Proof. According to Theorem 4, we have γ_st^k(G) +d^(j,k)_st (G)≤ jn

d^(j,k)_st (G)+d^(j,k)_st (G).

Theorem 5 implies that 1 ≤ d^(j,k)_st (G) ≤ jδ(G)/k ≤ jn. Using these inequalities, and the fact that the functiong(x) =x+jn/xis decreasing for 1≤x≤√

jnand increasing for√

jn≤x≤jn, we deduce that

γ^k_st(G) +d^(j,k)_st (G)≤max{jn+ 1,1 +jn}=jn+ 1, and the proof is complete.

Fork≥2 we will improve Corollary 6.

Theorem 7.Letk≥2 be an integer. IfGis a graph of ordern≥3 with minimum degreeδ(G)≥k, thenγ_st^k(G) +d^(j,k)_st (G)≤n+ 1 whenj= 1 and

γ_st^k(G) +d^(j,k)_st (G)≤ jn 2 + 2 whenj≥2.

Proof. Ifj= 1, then it is just the same as Corollary 6. Assume next thatj≥2.

Case 1: Assume thatγ_st^k(G) = 1. Then the conditionk≥2 and Theorem 5 lead to

γ_st^k(G) +d^(j,k)_st (G) = 1 +d^(j,k)_st (G)≤1 +jn k ≤ jn

2 + 2 as desired.

Case 2: Assume thatγ_st^k(G)≥2. According to Theorem 4, we have γ_st^k(G) +d^(j,k)_st (G)≤γ_st^k(G) + jn

γ_st^k(G).

The assumption yields to 2≤γ_st^k(G)≤n. Using the fact that the functiong(x) = x+jn/xis decreasing for 2≤x≤√

jnand increasing for√

jn≤x≤n, we observe that

γ^k_st(G) +d^(j,k)_st (G)≤max

2 +jn 2 , n+j

= nj 2 + 2, and the proof is complete.

For the special casej≤k we will improve Theorem 7.

Theorem 8. Letk andj be two integers such that 1≤j ≤k. If Gis a graph of ordern≥3 with minimum degreeδ(G)≥k, then

γ_st^k(G) +d^(j,k)_st (G)≤n+j.

Proof. Assume first thatγ_st^k(G)≤j. Then Theorem 5 leads to γ_st^k(G) +d^(j,k)_st (G) =j+j·δ(G)

k ≤j+δ(G)≤n+j.

Assume second thatγ^k_st(G)≥j. According to Theorem 4, we have γ_st^k(G) +d^(j,k)_st (G)≤γ_st^k(G) + jn

γ_st^k(G). (1)

The assumption yields toj≤γ_st^k(G)≤n. Using the fact that the function g(x) = x+jn/xis decreasing forj≤x≤√

jnand increasing for√

jn≤x≤n, we observe by (1) that

γ^k_st(G) +d^(j,k)_st (G)≤max{j+n, n+j}=n+j, and the proof is complete.

Theorem 9. If the graph G contains a vertex v such that d(v) ≤ k + 1, then d^(j,k)_st (G)≤j.

Proof. Let{f₁, f₂, . . . , f_d}be a signed total (j, k)-dominating family onGsuch that d=d^(j,k)_st (G). Since

x∈N(v)f_i(x)≥kand|N(v)| ≤k+ 1, we deduce thatf_i(x) = 1 for eachx∈N(v) and eachi∈ {1,2, . . . , d}. Ifxis an arbitrary neighbor ofv, then it follows that

d^(j,k)_st (G) =d=

d

i=1

f_i(x)≤j, and this is the desired upper bound.

Letj ≥ 1 be an integer, and letn = j+ 5. Now letF_n be a fan with vertex set{x₁, x₂, . . . , x_n}such thatx₁x₂. . . x_nx₁ is a cycle of lengthn andx_n is adjacent to x_i for each i= 2,3, . . . , n−2. For 3≤t ≤ n−3 define f_t : V(F_n) → {−1,1} by f_t(x_t) = −1 and f_t(x) = 1 for x ∈ V(F_n)\ {x_t}. Then it easy to see that {f₃, f₄, . . . , f_n−3}is a signed total (j,1)-dominating family onF_n. Therefore Theorem 9 implies thatd^(j,1)_st (F_j+5) =j.

This example demonstrates that Theorem 9 is sharp, at least fork= 1.

As an application of Theorem 5, we will prove the following Nordhaus-Gaddum type result.

Theorem 10. IfGis a graph of ordernsuch thatδ(G)≥kandδ(G)≥k, then d^(j,k)_st (G) +d^(j,k)_st (G)≤j(n−1)

k .

Moreover, ifd^(j,k)_st (G) +d^(j,k)_st (G) =^j(n−1)_k , thenGis regular.

Proof. Sinceδ(G)≥k andδ(G)≥k, it follows from Theorem 5 that d^(j,k)_st (G) +d^(j,k)_st (G) ≤ jδ(G)

k +jδ(G) k

= j

k(δ(G) +δ(G))

= j

k(δ(G) + (n−Δ(G)−1))

≤ j k(n−1),

and this is the desired Nordhaus-Gaddum inequality. IfGis not regular, then Δ(G)− δ(G) ≥ 1, and the above inequality chain leads to the better bound d^(j,k)_st (G) + d^(j,k)_st (G)≤ ^j(n−2)_k . This completes the proof.

Theorem 11.If vis a vertex of a graph Gsuch thatd(v) is odd andk is even or d(v) is even andk is odd, then

d^(j,k)_st (G)≤j·d(v) k+ 1 .

Proof. Let{f₁, f₂, . . . , f_d}be a signed total (j, k)-dominating family onGsuch that d=d^(j,k)_st (G). Assume first that d(v) is odd andk is even. The definition yields to

x∈N(v)f_i(x)≥k for eachi∈ {1,2, . . . , d}. On the left-hand side of this inequality a sum of an odd number of odd summands occurs. Therefore it is an odd number, and askis even, we obtain

x∈N(v)f_i(x)≥k+ 1 for eachi∈ {1,2, . . . , d}. It follows that

j·d(v) =

x∈N(v)

j≥

x∈N(v)

d

i=1

f_i(x)

=

d

i=1

x∈N(v)

f_i(x)

≥

d

i=1

(k+ 1) =d(k+ 1),

and this leads to the desired bound.

Assume next thatd(v) is even andkis odd. Note that

x∈N(v)f_i(x)≥kfor each i∈ {1,2, . . . , d}. On the left-hand side of this inequality a sum of an even number of odd summands occurs. Therefore it is an even number, and ask is odd, we obtain

x∈N(v)f_i(x) ≥k+ 1 for eachi∈ {1,2, . . . , d}. Now the desired bound follows as above, and the proof is complete.

The next result is an immediate consequence of Theorem 11.

Corollary 12. IfGis a graph such thatδ(G) is odd and kis even orδ(G) is even andk is odd, then

d^(j,k)_st (G)≤ j·δ(G) k+ 1 .

As an application of Corollary 12 we will improve the Nordhaus-Gaddum bound in Theorem 10 for some cases.

Theorem 13. LetGbe a graph of order n such that δ(G)≥ k andδ(G) ≥k. If Δ(G)−δ(G)≥1 orkis odd orkis even andδ(G) is odd ork,δ(G) andnare even, then

d^(j,k)_st (G) +d^(j,k)_st (G)<j(n−1)

k .

Proof. If Δ(G)−δ(G)≥1, then Theorem 10 implies the desired bound. Thus assume now thatGisδ(G)-regular.

Case 1: Assume thatk is odd. Ifδ(G) is even, then it follows from Theorem 5 and Corollary 12 that

d^(j,k)_st (G) +d^(j,k)_st (G) ≤ jδ(G)

k+ 1 +jδ(G) k

= jδ(G)

k+ 1 +j(n−δ(G)−1) k

< j(n−1)

k .

If δ(G) is odd, then n is even and thus δ(G) = n−δ(G)−1 is even. Combining Theorem 5 and Corollary 12, we find that

d^(j,k)_st (G) +d^(j,k)_st (G) ≤ jδ(G)

k +jδ(G) k+ 1

= j(n−δ(G)−1)

k +jδ(G)

k+ 1

< j(n−1)

k ,

and this completes the proof of Case 1.

Case 2: Assume thatk is even. Ifδ(G) is odd, then it follows from Theorem 5 and Corollary 12 that

d^(j,k)_st (G) +d^(j,k)_st (G)≤ jδ(G)

k+ 1 +jδ(G)

k < j(n−1)

k .

Ifδ(G) andnare even, thenδ(G) =n−δ(G)−1 is odd, and we obtain the desired bound as above.

Theorem 14. IfG is a graph such thatj is odd andd^(j,k)_st (G) is even or j is even andd^(j,k)_st (G) is odd, then

d^(j,k)_st (G)≤(j−1)δ(G)

k .

Proof. Let{f₁, f₂, . . . , f_d}be a signed total (j, k)-dominating family onGsuch that d=d^(j,k)_st (G). Assume first thatjis odd anddis even. Ifx∈V(G) is an arbitrary vertex, then d

i=1f_i(x) ≤ j. On the left-hand side of this inequality a sum of an even number of odd summands occurs. Therefore it is an even number, and asjis odd, we obtain_d

i=1f_i(x)≤j−1 for eachx∈V(G). Ifvis a vertex of minimum degree, then it follows that

d·k =

d

i=1

k≤^d

i=1

x∈N(v)

f_i(x)

=

x∈N(v)

d

i=1

f_i(x)

≤

x∈N(v)

(j−1)

= (j−1)δ(G),

and this yields to the desired bound. Assume second thatjis even anddis odd. If x∈V(G) is an arbitrary vertex, then_d

i=1f_i(x)≤j. On the left-hand side of this inequality a sum of an odd number of odd summands occurs. Therefore it is an odd number, and asjis even, we obtain_d

i=1f_i(x)≤j−1 for eachx∈V(G). Now the desired bound follows as above, and the proof is complete.

If we suppose in the casej= 1 thatd^(1,k)_st (G) is an even integer, then Theorem 14 leads to the contradictiond^(1,k)_st (G)≤0. Consequently, we obtain the next known result.

Corollary 15. ([5]) The signed total (1, k)-domatic number d^(1,k)_st (G) is an odd integer.

The special casek= 1 in Corollary 15 can be found in [4].

Theorem 16.Letj≥2 andk≥1 be integers, and letGbe a graph with minimum degreeδ(G)≥k. Thend^(j,k)_st (G) = 1 if and only if for every vertexv∈V(G), there exists a vertexu∈N(v) such thatk≤d(u)≤k+ 1.

Proof. Assume that for every vertexv∈V(G), there exists a vertexu∈N(v) such thatk≤d(u)≤k+ 1. Observation 1 implies thatγ_st^k(G) =n and thus we deduce from Observation 2 thatd^(j,k)_st (G) = 1.

Conversely, assume thatd^(j,k)_st (G) = 1. Suppose to the contrary thatGcontains a vertexwsuch d(x) ≥k+ 2 for eachx∈N(w). Then the functionsf_i :V(G) → {−1,1} such thatf₁(x) = 1 for eachx∈V(G) and f₂(w) =−1 and f₂(x) = 1 for each vertexx∈V(G)\ {w}are signed totalk-dominating functions onGsuch that f₁(x) +f₂(x) ≤ 2 ≤ j for each vertex x ∈ V(G). Thus {f₁, f₂} is a signed total (j, k)-dominating family onG, a contradiction tod^(j,k)_st (G) = 1.

Now we present a lower bound on the signed total (j, k)-domatic number.

Theorem 17.Letj, k≥1 be integers such thatj≤k+ 2, and letGbe a graph with minimum degreeδ(G)≥k. IfGcontains a vertexv∈V(G) such that all vertices of N[N[v]] have degree at leastk+ 2, thend^(j,k)_st (G)≥j.

Proof. Let{u₁, u₂, . . . , u_j} ⊆N(v). The hypothesis that all vertices ofN[N[v]] have degree at least k+ 2 implies that the functions f_i : V(G) → {−1,1} such that f_i(u_i) = −1 and f_i(x) = 1 for each vertex x ∈ V(G)\ {u_i} are signed total k- dominating functions onGfori∈ {1,2, . . . , j}. Sincef₁(x) +f₂(x) +. . .+f_j(x)≤j for each vertex x ∈V(G), we observe that {f₁, f₂, . . . , f_j} is a signed total (j, k)- dominating family onG, and Theorem 17 is proved.

Corollary 18. Letj, k ≥ 1 be integers such thatj ≤ k+ 2. If G is a graph of minimum degreeδ(G)≥k+ 2, thend^(j,k)_st (G)≥j.

Next we determime the signed total (j, k)-domatic number for some families of graphs.

Theorem 19. Letj, k≥1 be integers, and letGbe a graph with minimum degree δ(G)≥k+ 2.

1. If δ(G) =k+ 2t+ 1 with an integert≥1 and j <min_k+1

t ,^2k+2t+1_2t+1 , then d^(j,k)_st (G) =j.

2. Ifδ(G) =k+ 2twith an integert≥1 andj <^k

t, thend^(j,k)_st (G) =j.

Proof. 1. Letd=d^(j,k)_st (G), and letδ(G) =k+ 2t+ 1 with an integert≥1. Assume thatj <min_k+1

t ,^2k+2t+1_2t+1

. Sincekandδ(G) are of different parity, we deduce from Corollary 12 that

d≤ jδ(G)

k+ 1 =j(k+ 2t+ 1) k+ 1 . Asj <^k+1_t this leads to

d≤ j(k+ 2t+ 1)

k+ 1 < j+ 2

and sod≤j+ 1. If we suppose thatd=j+ 1, then we observe thatd andjare of different parity. Using the conditionj < ^2k+2t+1_2t+1 and Theorem 14, we arrive at the contradiction

j+ 1 =d≤j−1

k (k+ 2t+ 1)< j+ 1.

Therefored≤j, and Corollary 18 yields to the desired resultd=j.

2. Letd=d^(j,k)_st (G), and letδ(G) =k+ 2twith an integer t≥1. Assume that j <^k_t. It follows from Theorem 5 and the conditionj <^k_t that

d≤jδ(G)

k =j(k+ 2t)

k < j+ 2

and sod≤j+ 1. If we suppose thatd=j+ 1, thendandj are of different parity.

Applying Theorem 14, we obtain the contradiction j+ 1 =d≤ j−1

k (k+ 2t)< j+ 1.

Therefored≤j, and Corollary 18 yields to the desired resultd=j.

Theorem 19 demonstrates that the bound in Corollary 18 is sharp. Finally, we present a supplement to Theorems 10 and 13.

Theorem 20.Letk≥2 be an even integer, and letGbe aδ(G)-graph of odd order nsuch thatδ(G)≥k andδ(G)≥k,δ(G) is even andδ(G) =k+ 2twith an integer t≥1. Ifj < k/t, then

d^(j,k)_st (G) +d^(j,k)_st (G)≤j(n−2t−1)

k <j(n−1)

k .

Proof. SinceGis (n−k−2t−1)-regular andj < k/t, we conclude from Theorems 5 and 19 that

d^(j,k)_st (G) +d^(j,k)_st (G) ≤ j+jδ(G) k

= j+j(n−k−2t−1) k

= j(n−2t−1)

k <j(n−1)

k .

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(Received 27 Mar 2012; revised 11 June 2013)