Day 07Part 04 – Discrete Mathematics for Gate 2025 Bipartite graph and it’s Concept.<\/p>\n
[fvplayer id=”165″]<\/p>\n
Here\u2019s a clear and concise explanation of the Bipartite Graph<\/strong> concept from Discrete Mathematics<\/strong>, tailored for GATE 2025 preparation (Day 07, Part 04)<\/strong>:<\/p>\n A bipartite graph<\/strong> is a type of graph where:<\/p>\n Vertices can be divided into two disjoint sets<\/strong>: Every edge connects a vertex from set UU<\/span>U<\/span><\/span><\/span><\/span> to a vertex in set VV<\/span>V<\/span><\/span><\/span><\/span><\/strong> A graph G=(V,E)G = (V, E)<\/span>G<\/span>=<\/span><\/span>(<\/span>V<\/span>,<\/span>E<\/span>)<\/span><\/span><\/span><\/span> is bipartite<\/strong> if: A graph is bipartite if and only if it does not contain any odd-length cycles<\/strong>.<\/p>\n<\/li>\n Bipartite graphs are 2-colorable<\/strong>, meaning you can color vertices with two colors such that no two adjacent vertices share the same color.<\/p>\n<\/li>\n<\/ul>\n Use Breadth-First Search (BFS)<\/strong> or Depth-First Search (DFS)<\/strong>:<\/p>\n Try to color the graph using two colors.<\/p>\n<\/li>\n If at any point, two adjacent vertices have the same color \u2192 not bipartite<\/strong>.<\/p>\n<\/li>\n<\/ul>\n Let\u2019s take a graph with vertices: Partition: All edges go from V1\u2192V2V_1 \\to V_2<\/span>V<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>\u2192<\/span><\/span>V<\/span>
\n\ud83d\udcd8 Day 07 Part 04 \u2013 Discrete Mathematics for GATE 2025<\/strong><\/h2>\n
\ud83e\uddee Topic: Bipartite Graph and Its Concept<\/strong><\/h3>\n
\n\u2705 What is a Bipartite Graph?<\/strong><\/h3>\n
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UU<\/span>U<\/span><\/span><\/span><\/span> and VV<\/span>V<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n
(i.e., no edge exists between vertices within the same set<\/strong>)<\/p>\n<\/li>\n<\/ul>\n
\n\ud83e\udde9 Formal Definition:<\/strong><\/h3>\n
There exists a partition V=V1\u222aV2V = V_1 \\cup V_2<\/span>V<\/span>=<\/span><\/span>V<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>\u222a<\/span><\/span>V<\/span>2<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> such that for every edge (u,v)\u2208E(u, v) \\in E<\/span>(<\/span>u<\/span>,<\/span>v<\/span>)<\/span>\u2208<\/span><\/span>E<\/span><\/span><\/span><\/span>,
either u\u2208V1u \\in V_1<\/span>u<\/span>\u2208<\/span><\/span>V<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> and v\u2208V2v \\in V_2<\/span>v<\/span>\u2208<\/span><\/span>V<\/span>2<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, or vice versa.<\/p>\n
\n\ud83c\udfaf Key Properties:<\/strong><\/h3>\n
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\n\ud83e\udde0 How to Check if a Graph is Bipartite (GATE Tip):<\/strong><\/h3>\n
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\n\ud83e\uddee Example:<\/strong><\/h3>\n
V={A,B,C,D}V = \\{A, B, C, D\\}<\/span>V<\/span>=<\/span><\/span>{<\/span>A<\/span>,<\/span>B<\/span>,<\/span>C<\/span>,<\/span>D<\/span>}<\/span><\/span><\/span><\/span> and edges:
E={(A,B),(A,D),(C,B),(C,D)}E = \\{(A,B), (A,D), (C,B), (C,D)\\}<\/span>E<\/span>=<\/span><\/span>{(<\/span>A<\/span>,<\/span>B<\/span>)<\/span>,<\/span>(<\/span>A<\/span>,<\/span>D<\/span>)<\/span>,<\/span>(<\/span>C<\/span>,<\/span>B<\/span>)<\/span>,<\/span>(<\/span>C<\/span>,<\/span>D<\/span>)}<\/span><\/span><\/span><\/span><\/p>\n
V1={A,C}V_1 = \\{A, C\\}<\/span>V<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>{<\/span>A<\/span>,<\/span>C<\/span>}<\/span><\/span><\/span><\/span>, V2={B,D}V_2 = \\{B, D\\}<\/span>V<\/span>2<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>{<\/span>B<\/span>,<\/span>D<\/span>}<\/span><\/span><\/span><\/span><\/p>\n