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 […]

Intereting Posts

Find the distance covered by the ant
Hermite polynomials recurrence relation
References for Banach Space Theory
Does this integral have a closed form or asymptotic expansion? $\int_0^\infty \frac{\sin(\beta u)}{1+u^\alpha} du$
Showing $a^2 < b^2$, if $0 < a < b$
What are the common abbreviation for minimum in equations?
Is $[0, 1) \times (0, 1)$ homeomorphic to $(0, 1) × (0, 1)$?
homotopic maps from the sphere to the sphere
Proving $ \binom n 0 ^2 + \binom n 1 ^2 + \dots + \binom n n ^2 = \binom { 2n} n $ without induction
Probability to choose specific item in a “weighted sampling without replacement” experiment
An elegant description for graded-module morphisms with non-zero zero component
How should I understand “$A$ unless $B$”?
Different version of Gauss's Lemma
Compute the limit $\sum_{n=1}^{\infty} \frac{n}{2^n}$
What is the difference between the Frobenius norm and the 2-norm of a matrix?