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E/ otherwise. G/ ! v/ otherwise. It then follows by properties (1)–(3) of the vertex coloring c0 that c0 is vertexdistinguishing. 4/ in Fig. 5. Next, suppose that r 3 is odd, say r D 2` C 1 for some positive integer `. We now construct an ordered list L of the n vertices of G, separated into r blocks B1 ; B2 ; : : : ; Br of k vertices each. 8). 9). 10). Then relabel the vertices of L as u1 ; u2 ; : : : ; un . We now construct a spanning subgraph H of G as in the case when r is even. That is, for integers i and j with 1 Ä i < j Ä n, the vertex ui is adjacent to uj in H if i C j Ä n C 1 and ui and uj do not belong to the same partite set of G.

G/ 2, which, of course, we 3n 2m already knew. G/ , while if k D 4 in 3 4n 2m 2n m 3n 2m D . G/. 17 ([24]). n C 2/=3. u1 ; u2 ; : : : ; u4q 1 / be a path of order 4q 1. u1 ; v2 ; w2 / of length 2 at u1 . ui ; viC1 ; wiC1 / of length 2 at ui . u4q 1 ; v4qC1 ; w4qC1 / of length 2 at u4q 1 . Denote the resulting tree by Tq , which has order 12q C 1. The tree T2 is shown in Fig. 12. 12q C 3/=3 D 4q C 1. Tq / Ä 4q C 1. Tq / ! ui viC1 / D 4q C 1 for 1 Ä i Ä 4q 1. ui uiC1 / D ˘ i 2q C 1 C 2 for 2q C 2 Ä i Ä 4q 2.

G/ D 4q. While the results presented on irregularity strength have either dealt with formulas for the irregularity strength of certain classes of graphs or lower bounds, we now present a number of upper bounds. Since the proofs for these results are lengthy and do not provide additional insight into this topic, such results will be stated without proofs. 20 ([3]). G/ Ä n 1. 20 is sharp. Because the star of order n is the only tree whose irregularity strength is n 1, there is an improved upper bound for other trees.

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