О смене знака числа доминирования границы графа Текст научной статьи по специальности «Математика»

Baogen Xu , Jinfeng Zhao

Y^K 519.17

ON REVERSE MINUS EDGE DOMINATION NUMBERS OF GRAPHS

1 Introduction

We use Bondy and Murty [1] for terminology and notation not defined here and consider simple graphs only.

Let G = (V, E) be a graph with the vertex set V = V(G) and the edge set E = E(G). For any vertex u e V(G), dG (u) denotes the degree of u in G , and NG (u) and NG[u] denote the open and closed neighborhood of u in G, resp., S = S(G)and A = A(G) denote the minimum degree and the maximal degree respectively. For any edge e = uv e E(G), NG (e) denotes the open edge-

neighborhood of e in G and NG[e] = NG(e) U {e} for the closed one. dG (e) = |Ng (e)| is called the degree of e in G . Clearly, dG (e) = dG (u) + dG (v) - 2 (e = uv) .If the graph is clear from the context, dG (u), Ng (u), Ng [u], Ng (e), Ng [e] can simply be denoted by d(u) , N(u), N[u], N(e) and N[e], respectively.

In recent years, several kinds of domination problems in graphs have been investigated [2-5].

Most of them belong to the vertex domination of graphs, such as signed domination, minus domination, majority domination, etc. Recently, the problem has been changed from vertex domination to edge domination, and a few results have been obtained about the edge domination of graphs, such as signed edge domination [6], signed star domination [7], minus edge domination [8], etc. The concept of reverse minus edge domination was introduced in [9] but there are no results that have been obtained about it. In this paper, we will obtain some bounds ofy'm (G) .

Definition 1[8] Let G = (V, E) be a graph, a function f : E ^ {-1,0,1} is said to be the minus edge dominating function (MEDF) of G if

Supported by Natural Science Foundation of P.R. China (10661007)

]f (e') —1 holds for every e e E(G) .The minus edge domination number y'm (G) of G is defined as

y'm (G) = mm^^ f(e)\ f is anMEDF of G} .

For any empty graph Kn, define y'm (Kn) = 0 . Definition 2[9] Let G = (V, E) be a graph, a function f : E ^ {-1,0,1} is said to be the reverse minus edge dominating function (RMEDF) of G if ]f (e') - 0 holds for every e e E(G) .The reverse minus edge domination number y'm (G) of G is defined as

y'm (G) = max{ ^^ f (e)\f is a RMEDF of G} . For any empty graph Kn , then define

ym (Kn)=0.

By the above definition, we have fm (G1 U G2) = fm (Gl) + fm (G2 ) for any two disjoint graphs G and G .

In this paper, we will give some bounds of y'm (G) for general graphs in terms of its order, maximum degree and minimum degree of G. In addition, we determine the exact value of y'm (G) for the regular

graphs, the wheel graphs and the complete bipartite graphs.

2 Some bounds for y'm (G)

Theorem 1 For any graph of order n, then

ym (G) - 0.

Proof: We define a function f as follows: For any edge e e E(G), f (e) = 0 .Note that f (N[e]) = 0, it is obvious that f is a reverse minus edge dominating function of G and ^ f (e) = 0,

eeE (G)

therefore y'm (G) — ^ f (e) = 0. We have com-

eeE (G)

иркутским государственный университет путей сообщения

pleted the proof of theorem 1.

#

Theorem 2 For any graph of order n (n > 4),

then

(g) < (»z^ +,, 8

where A = A(G) is the maximum degree ofG .

Proof: Let f be such a reverse minus edge dominating function of G that y'm (G) = X f (e) . De-

eeE (G )

fine A = {e e E(G)\ f (e) = 1}, B = {e e E(G)\ f (e) = -1}, С = {e e E(G)\ f (e) = 0}, .Obviously, f'm (G) = s -1.

IA=s, B|=t

We define a spanning subgraph H of G as fol-

lows:

V (H) = V(G), E(H) = A

Then

XdH (e) = X(dH (u) + dH (v) - 2) = X(^ (u))2 - 2s

eeA uveA ueV (G)

> ■

1

(

n

X dH (u)

^ueV(G) J

- 2s =

4s2

n

- 2s

n

n n

8s

2Д- 2 > da (eQ), thus 2Д- 2 >--3 , it implies

n

(2 A + 1)n , ,

s <-, and we have

8

, ^ n 4 _

fm (G) = s -1 < s--+1 = s--+1

n n

^(2A + 1)n n - 4 ^_(2A + 1)(n - 4)

8 n 8 .

We have completed the proof of theorem 2.

Theorem 3 For any graph G of order n, then

/m (G) <

n 2 (Д-J)

4(2A-1)

Where A = A(G) and d = d(G) are the maximum and minimum degree ofG .

Proof: Let f be such a reverse minus edge dominating function of G that y'm (G) = ^ f (e) . De-

eeE (G )

fine A = {e e E(G)\ f(e) = 1}, B = {e e E(G)\ f (e) = -1}, C = {e e E(G)\ f (e) = 0}.

We define two spanning subgraphs G and G of G as follows:

V(Gl) = V (G) V(G2) = V (G) EG) = A E(G2) = B For any vertex u e V(G), define d * (u) = dG (u) - da2 (u), it is obvious that

f

/ m

(G) = |E(Gj) - \E(G2)\ =1 Xd(u),

2 ueV(G)

(1)

It implies that there exists at least one edge e0 4S

in A so that (^ ) >--2, by the definition of

n

reverse minus edge domination, there exists at least 4S

--1 edges of B in N[e0 ] so that

n

4s , ,

t >--1, then

n

, . . 4s „ 4s , 8s ^ da (e0) >--2 +--1 =--3 , note that

then X d *(u) = 2f'm (G)

ueV (G)

For any edge e = uv e E(G), by the definition 2, we know that X f (e') < 0, note that the defini-

e'eN[e]

tion of d (u), then

d* (u) + d* (v) - f (uv) = X f (e') < 0, therefore

eeN[e]

X (d* (u) + d*(v) - f (uv)) < 0, it is

uveE (G)

X (d * (u) + d\v)) - X f (uv) < 0, thus

uveE (G) uveE (G)

XdG(u)d\u) <r'm(G) (2)

ueV (G)

Let X = {u e V(G)|d * (u) > 1},

Y = {u eV(G)|d * (u) < 0}, By (2), we have

AX d \u) + SX d \u) <fm (G),

ueY ueX

Thus A X d\u) <fm (G) + (A-5) X d*(u) (3)

ueV (G) ueX

We know that any vertex in X is not adjacent to another (otherwise, for any e = uv e E(G) ,

X f (e') = d*(u) + d*(v) - f (uv) > 1, a contradic-

eeN[e]

#

системным анализ и его приложения

tion), so for any vertex in X, 1 — d*(u) — n — |x| ,

n

therefore £ d* (u) — |X|(n — |X|) — —, combining

ueX 4

with (1) and (3), we have

.2

, it is

2Ay'm(G) <y'm(G) + (A-S)

n 4

ym (G) <

n2 (A -S)

4(2 A — 1)

of theorem 3. #

By Theorem 1 and 3, we have the following corollary immediately:

Theorem 4 For any regular graph G of order n , thenym(G) = 0 .

3 Some special classes of graphs

In this section, we determine y'm (G) for the

wheel graphs and the complete bipartite graphs

Theorem 5 For any wheel W„+l = Cn vKl(n — 3)of ordern +1, then

rL(W+1) = 0.

Proof: Let G = Wn+1, v (g) = ^, vx, • • •, vn}, where v0 is the center vertex, as i.n Fig. 1.

Fig .1. The graph G = Wr

n+1

Let f be such a reverse minus edge dominating function of G that y'm (G) = £ f (e) . Let Sn+1 be

eeE (G )

the star of order n +1 with center vertex v0 of G , and V(S^) = V(G), E(Sn+!) = {V0 Vi 1 — i — n},

Cn = Wn+r — v0. For simplicity, let

f(S„+r) = f(E(S„+!)), f(Cn) = f(E(Cn)).

By the definition of the reverse minus edge domination, we know that for any edge e e E(Sn+x),

f (N[e]) < 0, then £ f (N[e]) < 0, that is ,

esE ( Sn+i)

nf (Sn+l) + 2f (Cn) < 0.

For any edge e e E(Cn) , f (N[e]) < 0, then £ f (N[e]) < 0, it is 2f (Sn+i) + 3f (Cn) < 0.

eeE (Cn )

Case1:

If f (S„+i) < 0

then

.We have completed the proof

3f (Sn+i) + 3f (Cn) < 2f (Sn+i) + 3f (Cn) < 0,

therefore

ym (Wn+i) = f (Sn+l)+f (Cn) < 0.

Case2:

If f (Sn+i) >1

then

2f(Sn+l) + 2f(Cn) — nf (Sn+1) + 2f(Cn) — 0,

therefore

ym (W„+! ) = f (Sn+! ) + f (Cn) — 0. Combining with case 1 and case 2 we have y'm Wn+1) — 0, note that theorem 1, we have

y'm (Wn+1) — 0, thus y'm(Wn+!) = 0 .We have completed the proof of theorem 5. #

Theorem 6 For any complete bipartite graph

Kmn, thenym (Kmn ) = 0 .

Proof: Let Kmn = G = (V, E), f be such a reverse minus edge dominating function of G that

ym (G) = s f (e).

eeE (G )

For any edge e e E(G) , by the definition, we have f (N[e]) — 0, so £ f (N[e]) — 0, it is

eeE (G)

(m + n — 1) £ f(e) — 0, thusym (K^ ) — 0 .

eeE (G )

Combining with theorem 1, we have y'm (Kmn ) — 0 .Thus y'm (Km n) = 0 . We have completed the proof of theorem 6. #

REFERENCES

1. Bondy J. A., Murty V. S. R. Graph Theory with

Applications. Amsterdam : Elsevier, 1976.

2. A Note on the Lower Bounds of Signed Domina-

tion Number of a Graph / Zhang Z., Baogen Xu, Li Y., Liu L. // [J]. Discrete Math. 1999. № 195. P.295-298.

3. Baogen Xu, Cockayne E. J. Extremal Graphs for Inequalities Involving Domination Parameters // [J]. Discrete Math. 2000. № 216. P. 1-10.

V

V

иркутским государственный университет путей сообщения

4. Baogen Xu. On Minus Domination and Signed Domination in Graphs // [J]. J. Math. Res. & Ex-posotion. 2003. № 4 P. 586-590.

5. Cockayne E. J., Mynhart C. Y. On a Generalization

of Signed Domination Functions of Graphs // [J]. Ars. Combin. 1996. № 43. P. 235-245.

6. Baogen Xu. On Signed Edge Domination Numbers

of Graphs // [J]. Discrete Math. 2001. № 239 ( P. 179-189.

7. Baogen Xu. On Signed Star Domination Numbers

of Graphs // [J]. Journal of East China Jiaotong University. 2004. № 4 ( P. 116-118.

8. Baogen Xu, Shangchao Zhou. On Minus Edge Do-

mination of Graphs // [J]. Journal of Jiangxi Normal University (Natural Science). 2007. № 31 (1). P. 21-24.

9. Baogen Xu. Domination Theory in Graphs. [M].

Beijing : Science Press, 2008. P.5.

Er-gen Liu, Ke-wen Cai, Wu dan

UDC 519.17

ON THE GRACEFULNESS OF GRAPH C

8,7, n

1. Introduction

Posed by a number of circles graph is a kind of important and interesting graph, which gracefulness is on many scholars as the object of study [1~3]. The reference [4] is given graceful labeling of graph

1, 2n ' 2, 2n ' C6, 3, 2n . In this paper on the basis

of reference [4] to further expand, We proved that

graph C8Xn , C8An , C8,3,n and C8,4,n are the graceful

graph.

In this paper, our discussion is undirected simple graph. V(G) and E(G) are vertex set and edge set of graph G . Unspecified symbols and terminology are the same reference [5].

Definition 1.1 For G = (V, E), If for each one v eV, there exists a non-negative integer 0(v) (called the vertices v of the label), meet:

(1) Vu, v eV, if u * v, then 0(u) * 0(v);

(2) max0(v) \ v eV} =\E \;

(3) V^, e2 e E,

if

* e2 =

then

0'(e ) * 0'(e2), which 0'(e) =\ 0(u) - 0(v) \, e = uv.

Then G is called graceful graph. 0 is called graceful value or graceful labeling, 0' known as the edge of G induced value by 0.

Definition 1.2 Graph C8in denotes a graph

that is composed by n loops C8 which have i vertices adhered by order one after another, but from the second to 2n -1 circles , the adhered vertices, get 2 equidistant roads in each circle(not including the two circles adhered edge).

2. Main results

Theorem 2.1 C81n is a graceful graph.

Proof As the graph shown on fig. 1. Label all vertices as follows:

0(xu) = 0; 0(xl2) = 8n; 0(xu) = 1; 0(xu) = 8n -1; 0(x15) = 2 ; 0(x^6) = 8n - 3 ; 0(xyi) = 3; 0(x^8) = 8n - 4;

0(x) = 3i + 4 + 8(n - i), i = 2,3, 0(x 3) = 3i, i = 2,3, ■■■, n; 0(xiA ) = 3i + 5 + 8(n - i), i = 2,3, 0(x 5) = 3i -1, i = 2,3, ■■■, n; 0(x ) = 3i + 2 + 8(n - i), i = 2,3,

, n;

, n;

6(хг1) = 3i +1,7 = 2,3, •■■, n ; 0(x ) = 37 + 3 + 8(n - i),i = 2,3, •■■,

, n;

n.

Now we proof that the follow 0 is graceful labeling of C8Xn

(1) Easy to find from the above label, In the same circle, these labels are different and the value of the label type contains n is greater than non- n value. In different circles, the value of the label type including n decreases with i increasing; the value of the label type excluding n increases with i increasing. Therefore, All the vertices have different labels and | E(C8Xn) \= 8n = max0(xh,), x,, e V(CSXn).

ft

* Fund Project: Natural Science Foundation of Jiangxi Province (0611009), Scientific research projects of Education Office (GJJ08254).