Let $G = (V, E)$ be a bipartite graph with bipartition $V = A \sqcup B$. Assume that $\delta(G) \geq 1$, and that $\deg(a) \geq \deg(b)$ for every edge $\left\{a, b \right\} \in E $ with $a \in A$. Show that $G$ contains a matching of $A$, i.e. a matching that meets every vertex of […]

I’m struggling with this question, and I’ll be happy for a hint or a direction. Let G=($V_1\cup V_2$,E) with no isolated Vertices s.t. the average degree on $V_1$ is large than the average degree on $V_2$. Prove that there exists an edge $\{v_1,v_2\}\in E$ s.t. $v_i\in V_i$ and $deg(v_1)>deg(v_2)$. Thank you. Things I tried: Since […]

Let $A_{n,m}$ be the maximum number of edges that a bipartite graph with $n,m$ vertices can have when it doesn’t contain $4$-cycle. I have calculated some values: $A_{2,2}=3$, $A_{3,3}=6$, $A_{4,4}=9$, $A_{4,5}=10$, $A_{5,5}=12$. I am trying to find formula for $A_{n,m}$. Does anyone know it or a hint how to find? Especially I need the value […]

This is such a hard question to get my head around. Can anyone help solve this?

Let $K_{(m,n)}$ be the complete bipartite graph with $m$ and $n$ being the number of vertices in each partition. Is there an efficient way to list down or construct all its spanning subgraphs up to isomorphism? I tried finding the spanning subgraphs for small $m$ and $n$ and what I am doing is I start […]

My actual question is to find the number of transversal given a collection of set … After a little bit of study it has come down to: How can we count the number of matchings in a bipartite graph with parts of size $m$ and $n$ such that it covers all $m$ vertices of the […]

I have to prove that for a bipartite graph G on n vertices the number of edges in $G$ is at most $n^2/4$. I used induction on n. induction hypothesis:Suppose for a bipartite graph with less than n vertices the result holds true. Now take a bipartite graph on n vertices.Let $x,y$ be two vertices […]

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