## Homework Set VIIIa and Problem Set HH (3/17)

Do for discussion on Mon. 3/22:

## HH1(a), HH2(a).

Do for discussion on Wed. 3/24:

## HH1 and HH2: in each, do one or two of (b,c,d).

# HH3.

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## Problem Set HH

HH1. Use the proof method of Theorem 3.2.4 to decompose into subgraphs isomorphic to P_{3} :

- the Petersen graph, Fig. 1.1.13,
- the icosahedral graph I in Fig. 1.2.5,
- Fig. 2.3.4,
- the Tutte graph, Fig. 2.3.5.

HH2. Use the proof method of Theorem 3.1.7 (Listing's Theorem) to decompose into the smallest possible number of trails:

- the Petersen graph, Fig. 1.1.13,
- Fig. 2.3.6,
- Fig. 2.3.7,
- the Grötzsch graph, Fig. 2.1.6.

HH3. Do your best to give a correct proof that, in the infinite graph G shown here, each of the three lobes hanging off vertex x has a 1-way Eulerian trail. (All the lobes are the same, but I only drew a large part of one of them.)