Section 2 1 Vocabulary Euler Cycles CYCLE a

Section 2. 1 Vocabulary Euler Cycles CYCLE – a sequence of consecutively linked edges (x 1, x 2), (x 2, x 3), …, (xn -1, xn) whose starting vertex is the ending vertex (x 1 = xn) and in which no edge can appear more than once. No edge can appear more than once. EULER CYCLE – a cycle that contains all the edges in a graph (and visits each vertex at least once). TRAIL – a sequence of consecutively linked edges in which no edge can appear more than once. MULTIGRAPHS – a graph which permits several edges joining the same pair of vertices. Ex. a d b c

Example a Euler cycle: a, d, c, i, j, n, o, k, j, d, e, k, l, h, m, g, h, f, e, b, a b g f c d j (a) e h k i l n m o c d e g f h (b) i j k l m

EULER TRAIL – a trail that contains all the edges in a graph (and visits each vertex at least once). DEADHEADING EDGE – edges added to a graph. LINE GRAPH – The line graph L(G) of graph of G has a vertex for each edge of G, and two of these vertices are adjacent if and only if the corresponding edges in G have a common end vertex. 6 Ex. 2 5 3 1 3 2 1 5 6 G L(G)

Origin of The Euler Cycle The Seven Bridges of Konigsberg A B C D A C B D

Theorem An undirected multigraph has an Euler cycle if and only if it is connected and has all the vertices of even degree. PROOF: • Suppose a multigraph G satisfies the two given conditions. • Pick any vertex a and trace out a trail. • Let C by the cycle thus generated and let G’ be the multigraph consisting of the remaining edges G-C. a b f c d j e k i (G) g d e g f h h l n c o Red = cycle C m i j k (G’) l m

• Because the original path was connected, C and G’ must have a common vertex, call it a’. • Build a new cycle C’ tracing through G’ from a’’ just as C was traced in G from a. • Incorporate C’ into the cycle C at a’ to obtain a new and larger cycle C’. • Repeat this process by tracing a cycle in the graph of G’’ of still remaining edges and incorporate the cycle into C’ to obtain C’’. • Continue until no edges remain. • It must next be shown that if a Euler cycle exists, then the graph is connected and has all vertices of even degree. ( ) In the essence of time we will save this for a b another day. c d e g f c h i j Green = cycle C’ Blue = cycle C’’ k l e d g h j m f k i l n o G = C + C’’ m

Corollary A multigraph has an Euler trail, but not and Euler cycle, if and only if it is connected and has exactly two vertices of odd degree. PROOF: • Suppose a multigraph has an Euler trail T, but not an Euler cycle. • The starting and ending vertices of T must have odd degree while all other vertices have an even degree. Also the graph must be connected. Ex.

• Suppose on the other hand that a multigraph G is connected and has exactly two vertices, p and q, of odd degree. • Add a supplementary edge (p, q) to G to obtain a graph G’. p q • G’ now has all vertices of even degree and therefore has an Euler cycle (Euler cycle theorem). • Remove the edge (p, q) from C. • Without this edge C the Euler cycle is reduced to an Euler trail that includes all of edges of G. Hence the corollary.

K 2 K 6 K 2: Euler trail – Yes Euler cycle – No K 3: Euler trail – No Euler cycle – Yes K 4: Euler trail – No K 3 Euler cycle – No K 8 K 5: Euler trail – No Euler cycle – Yes K 6: Euler trail – No Euler cycle – No K 4 . . . K 8: Euler trail – No Euler cycle – No K 5

Class Exercises 1. A) Can a graph with an Euler cycle have a bridge (an edge whose removal disconnects the graph)? 2. B) Give an example of a 10 -edge graph with an Euler trail that has a bridge. Answers: A) No. If you start on one side of the bridge and then you cross it the only way you can get back to where you started is to cross the bridge again. This means you have to travel the same edge twice and thus, a Euler cycle cannot exist. B)

2. Find a minimal set of edges in the graph below whose removal produces an Euler cycle. All vertices are of even degree but we cannot find the Euler cycle. a f e b j g i d h c