An Euler tour of a strongly connected, directed graph G =...

An Euler tour of a strongly connected, directed graph $G = (V, E)$ is a cycle that traverses each edge of $G$ exactly once, although it may visit a vertex more than once. a. Show that $G$ has an Euler tour if and only if in-degree$(v)$ = out-degree$(v)$ for each vertex $v \in V$. b. Describe an $O(E)$-time algorithm to find an Euler tour of $G$ if one exists. (Hint: Merge edge-disjoint cycles.)

Transcript

00:01 Alright, so we have a given a graph g.

00:05 Let's draw a basic little graph g here.

00:11 Say something like this.

00:15 I'm going to label the vertices a, b, c, d, and the edges 1, 2, 3, 4, and 5.

00:30 We define the line graph, this is g, l of g, which has vertices that are the edges of g.

00:43 So it's going to be, the vertices now are 1, 2, 3, 4, and 5.

00:51 And the edges are, there's an edge between two of these vertices if the edges in g are connected to the vertex.

01:01 So in this example here, we see that at vertex a, 1, 2, and 3 all meet.

01:09 So 1, 2, and 3 should be part of a complete subgraph here.

01:13 At vertex b, 1 and 4 meet.

01:16 At c, 2, 4, and 5 meet.

01:21 So we want 2 and 4 to meet, 2 and 5 as well as 4 and 5.

01:27 And then here at d, 3 and 5 meet.

01:30 So that should be like that.

01:35 Because i like drawing planar graphs as part of planes, i'm actually going to move 3 out here so i can draw that.

01:47 So that's the line graph.

01:50 We want to show that if g has an euler cycle, then so does l of g.

01:56 That is to say, if g is eulerian, then l of g is eulerian.

02:01 Let's go ahead and do that real quick.

02:03 Let's say that g is eulerian.

02:05 G, euler.

02:09 What that means is that v of g, excuse me, any vertex for all vertices v and g, in the vertex set, we have that the degree of v is even.

02:28 So we'll call it 2k.

02:31 That's great.

02:34 Let's take an edge, uv, in the edge set of g.

02:41 And we want to consider that, of course, as a vertex in the line graph.

02:47 So it's in the vertex set of the line graph.

02:55 What is the degree of this? so the degree of uv is going to be, i'm going to claim that it is going to be the degree of u minus 1 plus the degree of v minus 1.

03:16 Why is that? well, the degree of an edge is going to be all of the edges that, here, v, u here.

03:27 The degree of this edge is going to be all of the edges that it connects to somewhere.

03:33 So that's going to be, if you look at this here, these edges are going to go off somewhere...

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