Graceful, Harmonious and Magic Type Labelings by Susana C. López & Francesc A. Muntaner-Batle

Author:Susana C. López & Francesc A. Muntaner-Batle
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Figures 5.7 and 5.8 illustrate examples of the previous definitions.

Fig. 5.7A tree with an even degree sequence and even diameter

Fig. 5.8A tree with a quasi even degree sequence and odd diameter

Theorem 5.4 (​​[10])

Every tree with an even or quasi even degree sequence is graceful.

In order to prove Theorem 5.4, we introduce the following results. The first one establishes a relation between the size and the diameter of a tree with a (quasi) even degree sequence.

Lemma 5.3 (​​[10])

Let T be a tree of size m and even diameter D. If T has an even degree sequence, then m + D∕2 is even. If T has a quasi even degree sequence, then m + D∕2 is odd.

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