By Chartrand G., Lesniak L., Zhang P.

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B) What is the fewest number of edges needed to proceed from u to v along edges of G? (c) Starting with v, say, is it possible to proceed along all the edges (or about all the vertices) of G without repeating any and returning to the starting vertex? For the purpose of studying questions such as these, it is useful to introduce some new concepts. 3. CONNECTED GRAPHS AND DISTANCE ✐ 31 Walks, Trails and Paths For two (not necessarily distinct) vertices u and v in a graph G, a u − v walk W in G is a sequence of vertices in G, beginning with u and ending at v such that consecutive vertices in W are adjacent in G.

Continuing in this manner, we get s′2 : 1, 1, 2, 2, 2, 1, 1, 1, 1 s2 : 2, 2, 2, 1, 1, 1, 1, 1, 1 s′3 = s3 : 1, 1, 1, 1, 1, 1, 1, 1 s′4 : 0, 1, 1, 1, 1, 1, 1, 1 s4 : 1, 1, 1, 1, 1, 1, 0 s′5 : 0, 1, 1, 1, 1, 0 s5 : 1, 1, 1, 1, 0, 0 s′6 : 0, 1, 1, 0, 0 s6 : 1, 1, 0, 0, 0 s′7 = s7 : 0, 0, 0, 0 Therefore, s is graphical. 12 that s is graphical. 29. 12, each of the sequences s2 , s1 and s is graphical. To construct a graph with degree sequence s2 , we proceed in reverse from s′3 = s3 to s2 , observing that a vertex should be added to G3 so that it is adjacent to two vertices of degree 1.

25: The graphs G and H in Exercise 9 10. (a) Give an example of two non-isomorphic regular graphs G1 and G2 of the same order and same size such that (1) for every two vertices v1 ∈ V (G1 ) and v2 ∈ V (G2 ), G1 − v1 ∼ = G2 − v2 and (2) there exist 2-element subsets S1 ⊆ V (G1 ) and S2 ⊆ V (G2 ) such that G1 − S1 ∼ = G2 − S2 . (b) Give an example of two non-isomorphic regular graphs H1 and H2 of the same order and same size such that (1) for every 2-element subsets S1 ⊆ V (H1 ) and S2 ⊆ V (H2 ), H1 − S1 ∼ = H2 − S2 and there exist 3-element subsets S1′ ⊆ V (H1 ) and S2′ ⊆ V (H2 ) such that H1 − S1′ ∼ = H2 − S2′ .