In this note we give a generalized version of Vizing's conjecture concerning the distance domination number for the cartesian product of two graphs

Mostafa Blidia

Mustapha Chellali

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Opuscula Mathematica  Vol. 27  No. 2  2007Mostafa Blidia, Mustapha ChellaliA NOTE ON VIZING'S GENERALIZED CONJECTUREAbstract. In this note we give a generalized version of Vizing's conjecture concerning thedistance domination number for the cartesian product of two graphs.Keywords: graph, dominating sets, Vizing's conjecture.Mathematics Subject Classication: 05C69.1. INTRODUCTIONLet G = (V;E) be a simple and nite graph. A set D  V is a dominating set of G ifeach vertex in V  D is adjacent to at least one vertex in D. The domination number(G) is the minimum cardinality of a dominating set of G. For further details ondomination in graphs, see the monographs by Haynes, Hedetniemi, and Slater [4,5].The cartesian product G  H of two graphs G;H is the graph with V (G  H) =V (G)  V (H) and two vertices (u;v) and (w;t) being adjacent in G  H if eitheru = w and vt 2 E(H) or uw 2 E(G) and v = t. In [11] Vizing conjectured thatfor any graphs G and H, (G  H)  (G)  (H). No positive or negative answerto Vizing's conjecture has been given yet. Jacobson and Kinch [6,7] gave importantresults to support the conjecture.The distance between two vertices x and y, denoted by dG(x;y), is the length ofa shortest path from x to y. If S is a set of vertices of G and v is a vertex of G,then the distance from v to S, denoted by dG(v;S), is the minimum distance fromv to a vertex of S. The open k-neighborhood Nk(x) of a vertex x of G is dened asNk(x) = fy 2 V : dG(x;y)  k;x 6= yg and the closed k-neighborhood Nk[x] of x isNk(x)[fxg. The maximum k-degree of G is given by k(G) = maxfjNk(v)j : v 2 V g.Meir and Moon [10] introduced distance k concepts, i.e., the k-domination and thek-packing concepts. Let k  1 be an integer, a set Dk  V is said to be k-dominating(resp. k-packing) set of G if for every vertex v of G, dG(v;Dk)  k (resp. dG(x;y) > kfor all pairs of distinct vertices x and y in Dk). The k-dominating number k(G) (resp.the k-packing number k(G)) is the size of a smallest k-dominating (resp. a largest181182 Mostafa Blidia, Mustapha Chellalik-packing) set of G. Note that 1(G) = (G), also every k-packing set of G is ak-dominating set of G, therefore k(G)  k(G).Let k  1 be an integer. For any graph G, the k-th power of G is the graph denotedGk with V (Gk) = V (G) and two vertices u and v are adjacent if 1  dG(u;v)  k.Remark 1. It is easy to show that for an arbitrary graph G, (Gk) = k(G) and2(Gk) = 2k(G).2. GENERALIZATION OF VIZING'S CONJECTUREConsider two graphs G and H. Since Gk Hk is a spanning graph of (GH)k, thereis (Gk  Hk)  ((G  H)k).a) b)   Fig. 1. P24  P23 (a); (P4  P3)2 (b)To see that the above inequality may be strict, consider the graphs shown inFigures 1a and 1b, where the domination numbers are 3 and 2, respectively.Therefore, even if we suppose that Vizing's conjecture is veried for any graphs Gand H, we cannot conclude directly that also the k-domination number of the productof any two graphs is equal to at least the product of the k-domination numbers ofthese graphs. Hence, we state below a conjecture which generalizes Vizing's.Vizing's Generalized Conjecture (VGC): for any graphs G and H,k(G  H)  k(G)k(H):An immediate consequence of this conjecture is the following result.Proposition 1. Let k  1 be an integer. If VGC is true for the product of G and H,then Vizing's conjecture is also true for the product of Gk and Hk.Proof. Since Gk Hk is a spanning graph of (GH)k, it follows that (Gk Hk) ((G  H)k) = k(G  H)  k(G)k(H) = (Gk)(Hk).The next proposition is easy to prove.A note on Vizing's generalized conjecture 183Proposition 2. Let k  1 be an integer. For any graphs G and H withminfk(G);k(H)g = 1;k(G  H)  k(G)k(H).Proposition 3. Let k  1 be an integer. For any graph G and its complement G,k(G  G)  k(G)k(G):Proof. The case k = 1 has been proved by Jaeger and Payan [8] as well as El-Zaharand Pareek [2]. Assume that k  2, and let S be a maximum k-packing set of G. IfjSj = 1, then k(G) = 1, and by Proposition 2, we obtain the result. If jSj  2, thenevery vertex v of V (G) is at a distance not longer than two from S in G. Thus, eachvertex of S k dominates G. Hence, k(G) = 1, and by Proposition 2, we obtain theresult.Our next results extend those of [6] and [7].Proposition 4. Let k  1 be an integer. For any graphs G and H,k(G  H)  maxjV (H)j1 + k(H)k(G);jV (G)j1 + k(G)k(H):Proof. Let D be a minimum k-dominating set of GH and let i be a vertex of V (H).We put:Gi = G  fig,Di = D \ Gi for i = 1;:::;jV (H)j, andFi = ft 2 V (Gi) : dGH(t;Di) > kg.Since the graphs Gi and G are isomorphic, k(Gi) = k(G). Then for every i,jFij  k(G)   jDij, as otherwise Fi [ Di is a k-dominating set of Gi of a size lessthan k(G), a contradiction. Consequently,k(H)jDj jV (H)j Pi=1jFij jV (H)j Pi=1( k(G)   jDij) = jV (H)j k(G)   jDj, and hencek(G  H) jV (H)j1 + k(H)k(G):The second part is obtained similarly.Proposition 5. Let k  1 be an integer. For any graphs G and H,k(G  H)  maxfk(G)2k(H);k(H)2k(G)g:Proof. Let D be a minimum k-dominating set of G  H. Clearly, for any vertexi 2 V (H), there is j(V (G)  Nk[i]) \ Dj  k(G).Now let W be the maximum 2k-packing set of H. Then for any pair of vertices iand j in W, Nk[i] \ Nk[j] = ;. It follows that (V (G)  Nk[i]) \ (V (G)  Nk[j]) = ;,implying thatjDj Xi2Wj(V (G)  Nk[i]) \ Dj Xi2Wk(G) = k(G)2k(H):Likewise, k(G  H)  k(H)2k(G).184 Mostafa Blidia, Mustapha ChellaliIn [10], Meir and Moon showed that every tree T satises k(T) = 2k(T). Thisequality is also true for strongly chordal graphs (which contain trees). Indeed, it isenough to combine the results of Lubiw and Farber with Remark 1. Lubiw [9] hasproved that if G is a strongly chordal graph, then for any positive integer k, Gk isstrongly chordal graph, while Farber [3] has established that (G) = 2(G). We notethat Domke, Hedetniemi and Laskar [1] have proved the equality between k and 2kfor block graphs which are contained in strongly chordal class.According to Proposition 5, the following holds true.Proposition 6. Let k  1 be an integer. If G is a strongly chordal graph, then forany graph H,k(G  H)  k(H)k(G):REFERENCES[1] G.S. Domke, S.T. Hedetniemi, R. Laskar, Generalized packings and coverings of graphs,Congr. Numer. 62 (1988), 258270.[2] M. El-Zahar, C.M. Pareek, Domination number of products of graphs, Ars Combin. 31(1991), 223227.[3] M. Farber, Domination, independant domination and duality in strongly chordal graphs,Discrete Appl. Math. 7 (1984), 115130.[4] T.W. Haynes, S.T. Hedetniemi, P.J. Slater, Fundamentals of Domination in Graphs,Marcel Dekker, New York, 1998.[5] T.W. Haynes, S.T. Hedetniemi, P.J. Slater (eds), Domination in Graphs: AdvancedTopics, Marcel Dekker, New York, 1998.[6] M.S. Jacobson, L.F. Kinch, On domination of the products of graphs (I), Ars Combin.18 (1983), 3444.[7] M.S. Jacobson, L.F. Kinch, On domination of the products of graphs (II), J. GraphTheory 10 (1986), 97106.[8] F. Jaeger, C. Payan, Relation de type Nordhauss-Gaddum pour le nombre d'absorptiond'un graphe simple, C.R. acad. SC. Paris, Serie A, 274 (1972), 728730.[9] A. Lubiw,  -Free matrices, Master thesis, Univ of Waterloo, Antario 1982.[10] A. Meir, J.W. Moon, Relations between packing and covering numbers of trees, PacicJ. Math. 61 (1975) 1.[11] V.G. Vizing, The cartesian product of graphs, Vyc. Sis 9 (1963), 3043.Mostafa Blidiamblidia@hotmail.comA note on Vizing's generalized conjecture 185University of BlidaDepartment of MathematicsB.P. 270, Blida, AlgeriaMustapha Chellalim_chellali@yahoo.comUniversity of BlidaDepartment of MathematicsB.P. 270, Blida, AlgeriaReceived: September 19, 2006.

A note on a Vizing's generalized conjecture

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