m+n

alternatives Again, after assigning one student a topic, we reduce this down to the previous case of two students liking only one topic. \renewcommand{\topfraction}{.8} \def\circleBlabel{(1.5,.6) node[above]{$B$}} We need one new definition: The distance between vertices $$v$$ and $$w$$, $$\d(v,w)$$, is the length of a shortest walk between the two. One way $$G$$ could not have a matching is if there is a vertex in $$A$$ not adjacent to any vertex in $$B$$ (so having degree 0). The only such graphs with Hamilton cycles are those in which $$m=n$$. |N(S)| \ge |S| \def\Z{\mathbb Z} Your goal is to find all the possible obstructions to a graph having a perfect matching. \def\X{\mathbb X} Write a careful proof of the matching condition above. Theorem – A simple graph is bipartite if and only if it is possible to assign one of two different colors to each vertex of the graph so that no two adjacent are assigned the same color.

alternatives Again, after assigning one student a topic, we reduce this down to the previous case of two students liking only one topic. \renewcommand{\topfraction}{.8} \def\circleBlabel{(1.5,.6) node[above]{$B$}} We need one new definition: The distance between vertices $$v$$ and $$w$$, $$\d(v,w)$$, is the length of a shortest walk between the two. One way $$G$$ could not have a matching is if there is a vertex in $$A$$ not adjacent to any vertex in $$B$$ (so having degree 0). The only such graphs with Hamilton cycles are those in which $$m=n$$. |N(S)| \ge |S| \def\Z{\mathbb Z} Your goal is to find all the possible obstructions to a graph having a perfect matching. \def\X{\mathbb X} Write a careful proof of the matching condition above. Theorem – A simple graph is bipartite if and only if it is possible to assign one of two different colors to each vertex of the graph so that no two adjacent are assigned the same color.