The irregularity strength and cost of the union of cliques

Assign positive integer weights to the edges of a simple graph with no component isomorphic to Ki or 1£2, in such a way that the graph becomes irregular, i.e., the weight sums at the vertices become pairwise distinct. The minimum of the largest weights assigned over all such irregular assignments on the vertex-disjoint union of complete graphs is determined. The method of proof also yields the smallest possible total increase in the sum of edge weights in irregular asignments, called irregularity cost. 1. I n t r o d u c t i o n Let G = ( V ( G ) , E ( G ) ) be a simple graph having no connected components isomorphic to Ki or /£2. A network G(w) consists of the underlying graph G together with an assignment w : E ( G ) ~ Z +. Sometimes we shall denote a network simply by G if the underlying graph is understood and we need not specify the assignment w. For an edge e of G, the positive integer w(e) is called the weight of e. The strength s (G(w)) of the network G(w) is s (G(w) ) = max{w(e) : e E E(G)}. For each vertex x of G(w), the weighted degree wt(x) is defined as the sum of the weights of the edges incident to x. A network G(w) is called irregular if its distinct vertices have distinct weighted degrees. The irregularity strength s(G) of the graph G * Corresponding author. E-mail: [email protected]. 0012-365X/96/$t5.00 @ 1996--Elsevier Science B.V. All rights reserved SSDI 0012-365X(95)00186-7 180 S. Jendrol et al. / Discrete Mathematics 150 (1996) 179-186 is defined to be s(G) = min{s(G(w)) : G(w) is irregular}. That is, the irregularity strength of a graph G is the smallest possible value of s(G(w)), taken over all irregular networks G(w) having G as their underlying graph. A sequence (dl,d2 . . . . . dp) of positive integers is called the weighted degree sequence of a network G(w) if the vertices of G can be labelled vl, v2 ..... Vp such that wt(vi) = di for every i, 1 ~i <~ p. The problem of studying the irregularity strength of graphs was proposed by Chartrand et al. in [2]. It turned out to be rather hard, even for graphs having quite a simple structure (see e.g. [1, 3-5, 7, 8], and [9] for a survey). The problem of determining the irregularity strength of the vertex-disjoint union [_JtiKpi of cliques (i.e., ti copies of the complete graph on Pi vertices for i : 1,2 . . . . ) was studied first by Faudree et al. in [3]. They determined s(tK4), proposed an algorithm giving an upper bound for s((.J tiKp, ), and posed the conjecture that for all t ~> 2 and p > 3, s(tKp)= r(pt + p 2 ) / ( p 1)] holds. This conjecture turned out to be false: Jendroi and Tkfi(: [7] have proved that for every t >i-2 and p > 3, the exact formula is [ ( t p+ p 1 ) / ( p 1 ) ] i f [ ~ ] = 0(mod 2), s(tKp) = I t n l [ ( tp+p 2)/(p 1)] i f [ 2 j -----l(mod 2). On the other hand, s(Kp) = 3 for every p~>3 (see [2]). Moreover, the irregularity strength of tK3, t ~> 2, does not follow from the previous formula; as shown by Faudree et al. [4], it is [(3t + 1 )/2] + 2 for t -3 (rood 4), s(tK3) = [(3t + 1)/2] + 1 otherwise. In the present paper we solve the general problem where the clique components of G need not have the same size. We derive an exact formula (computed by a recursive procedure) for counting the irregularity strength of [.J tiKp, for all finite sequences of positive integers t;/> 1 and pi >1 3. There is a closely related problem, proposed by Harary and Oellermann (private communication, 1990), asking for the smallest possible extra cost vf~Aw(e)1) of an irregular labelling w of G. Formally, we define the irregularity cost e(G) of G as e ( G ) = m a x l Z (w(e)l ):G(w)is irregular } . eEE(G) In the first paper dealing with this graph invariant, Tuza [10] proved that ~(G) = ~n 2 + o(n 2) holds for 'almost all' graphs on n vertices, and this asymptotic equation remains valid for a randomly chosen G even if the graph is assumed to be relatively sparse. (The S. Jendrol et al./ Discrete Mathematics 150 (1996) 179-186 181 number of edges should grow with n4/3f(n), where f ( n ) is any function tending to infinity with n.) In a more recent manuscript, Jacobson et al. [6] study the related invariant 2(]E(G)I + e(G)) that they call the 'irregularity sum' of G for some particular classes of graphs. The latter terminology is related to the fact that 21E(G) ] + 2e(G) is the smallest possible degree sum in the irregular assignments of G. The ideas presented in Section 3 are suitable for proving tight results on both the irregularity strength and cost, and in this way we can determine the exact value of e( U tiKp,) as well. In both cases, it turns out that a lower bound obtained in a fairly natural way is in fact the correct answer for the problems.