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2. For each of the following bipartite graphs, determine whether or not there exists a matching that covers X. If there is, then list all the edges in that matching. If not, then find the subset of X that fails the condition in Hall's Theorem (Theorem 4.6.3). (c) X1 X2 X3 M У1 Уз Y₁ x1 X2 31 32 Y2 X3 уз X4 Y4 Y4 X5 Y5 23 XA XXXXXX Y3 YA IS X6 Y5 Y6 16 Y6

2. For each of the following bipartite graphs, determine whether or not there exists a matching that covers X. If there is, then list all the edges in that matching. If not, then find the subset of X that fails the condition in Hall's Theorem (Theorem 4.6.3). (c) X1 X2 X3 M У1 Уз Y₁ x1 X2 31 32 Y2 X3 уз X4 Y4 Y4 X5 Y5 23 XA XXXXXX Y3 YA IS X6 Y5 Y6 16 Y6

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2. For each of the following bipartite graphs, determine whether or not there exists a matching that covers \(X\). If there is, then list all the edges in that matching. If not, then find the subset of \(X\) that fails the condition in Hall’s Theorem (Theorem 4.6.3). **Graph (a):** - Vertices in set \(X\): \(x_1, x_2, x_3\) - Vertices in set \(Y\): \(y_1, y_2, y_3, y_4\) **Edges:** - \(x_1\) is connected to \(y_1, y_2\) - \(x_2\) is connected to \(y_2, y_3\) - \(x_3\) is connected to \(y_3, y_4\) **Graph (b):** - Vertices in set \(X\): \(x_1, x_2, x_3, x_4, x_5, x_6\) - Vertices in set \(Y\): \(y_1, y_2, y_3, y_4, y_5, y_6\) **Edges:** - \(x_1\) is connected to \(y_1, y_4, y_6\) - \(x_2\) is connected to \(y_1, y_3, y_4\) - \(x_3\) is connected to \(y_2, y_5, y_6\) - \(x_4\) is connected to \(y_2, y_3\) - \(x_5\) is connected to \(y_3, y_6\) - \(x_6\) is connected to \(y_4, y_5\) **Graph (c):** - Vertices in set \(X\): \(x_1, x_2, x_3, x_4, x_5, x_6\) - Vertices in set \(Y\): \(y_1, y_2, y_3, y_4, y_5, y_6\) **Edges:** - \(x_1\) is connected to \(y_1, y_4
Transcribed Image Text:2. For each of the following bipartite graphs, determine whether or not there exists a matching that covers \(X\). If there is, then list all the edges in that matching. If not, then find the subset of \(X\) that fails the condition in Hall’s Theorem (Theorem 4.6.3). **Graph (a):** - Vertices in set \(X\): \(x_1, x_2, x_3\) - Vertices in set \(Y\): \(y_1, y_2, y_3, y_4\) **Edges:** - \(x_1\) is connected to \(y_1, y_2\) - \(x_2\) is connected to \(y_2, y_3\) - \(x_3\) is connected to \(y_3, y_4\) **Graph (b):** - Vertices in set \(X\): \(x_1, x_2, x_3, x_4, x_5, x_6\) - Vertices in set \(Y\): \(y_1, y_2, y_3, y_4, y_5, y_6\) **Edges:** - \(x_1\) is connected to \(y_1, y_4, y_6\) - \(x_2\) is connected to \(y_1, y_3, y_4\) - \(x_3\) is connected to \(y_2, y_5, y_6\) - \(x_4\) is connected to \(y_2, y_3\) - \(x_5\) is connected to \(y_3, y_6\) - \(x_6\) is connected to \(y_4, y_5\) **Graph (c):** - Vertices in set \(X\): \(x_1, x_2, x_3, x_4, x_5, x_6\) - Vertices in set \(Y\): \(y_1, y_2, y_3, y_4, y_5, y_6\) **Edges:** - \(x_1\) is connected to \(y_1, y_4
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