In particular, we say that the chromatic number of any bipartite Likewise, if the graph can be colored using twoĬolors, define as the red colored nodes and as the green Simply take the set and color it red and color the It is easy to see that any bipartite graph is two colorable and To the vertices of the graph so that every pair of adjacent vertices have different colors. I.e, it is possible to assign one of two different colors Often, we think of bipartite graphs as two colorable graphs. In other words, there can be no edges between vertices in or no edges between vertices in. It is an undirected graph, there is no path from to either).Īlso the set is not a connected component since the only possible path in from to has to pass throughĪ graph (may be directed or undirected) is bipartite iff the vertex set can be partitioned into twoĪny edge in the graph goes from a vertex in to a vertex in or vice-versa. I.e., if youĬhoose any pair of vertices in the set then there is a path between them that just involves the verticesīy the same token, the set is not a connected component. We say that a subset of vertices is a connected component of a graph. In the second island to the first “island”. Let us start with the following example over the set of vertices. We now discuss the concept of strongly connected components In other words, a cycle cannot repeat an edge. The reason is that the undirected edge and areĬonsidered the same edge for the undirected graph and is being traversed twice. Important Note is not considered a cycle in the undirected The direction of a cycle does not matter in an undirected graph. Note that is a cycle in this graph of length. Which is the same cycle as (the cycle has length 2).Ĭonsider the following undirected graph instead: We could have written the sameĮxamples of cycles in this graph include: The choice of a starting point in a cycle is arbitrary in the following sense. The length of a cycle is the number of edges in it. Where other than itself no vertex or edge can repeat. I place “path” under quotes because we just defined paths soįormally, a cycle is a sequence of nodes and edges of the form In the previous example there is a path from to butĪ cycle in a graph is a “path” from a vertex back to itself. This is not always true in a directed graph. Is undirected, we note that any path from to may be reversed to immediately yield a path from to. Note that even though the graph is undirected, the path itself is from to. In the second case, the path does not end in, rather it ends in. In the first case, (and ) are repeated vertices which is not allowed. The length of a path is the number of edges in it. No vertex is repeated, i.e, each vertex is visited at most once. Given an undirected graph, a path from a vertex to aĭistinct vertex is an alternating sequence of vertices We end after a finite but arbitrary number of steps. In other words, a walk over a digraph is simply a sequence of visits to nodes and arcs of the graph whereinĪ single step of the walk consists of taking an outgoing arc from the node vertex to visit a node. A walk over is a sequence of nodes and arcs:
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