Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [bipartite-graphs]

For questions about graphs for which the set of vertices can be divided into two disjoint subsets such that no edge of the graph joins two vertices from same subset.

A bipartite graph is a graph $G=(V,E)$ such that the vertex set $V$ can be divided into two disjoint subsets such that no edge of $G$ joins two vertices from same subset. That is, $V=V_1\cup V_2$ such that $V_1\cap V_2=\emptyset$ and for any $e=uv\in E$, one of vertices among $u$ and $v$ belongs to $V_1$ and another belongs to $V_2$. There is no edge $e=uv\in E$ such that both $u$ and $v$ belongs to $V_1$ or both belongs to $V_2$.

780 questions
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Proof a graph is bipartite if and only if it contains no odd cycles

How can we prove that a graph is bipartite if and only if all of its cycles have even order? Also, does this theorem have a common name? I found it in a maths Olympiad toolbox.
Asinomás
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4 answers

Prove that a $k$-regular bipartite graph has a perfect matching

Prove that a $k$-regular bipartite graph has a perfect matching by using Hall's theorem. Let $S$ be any subset of the left side of the graph. The only thing I know is the number of things leaving the subset is $|S|\times k$.
TheMathNoob
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If $G$ is simple with $n$ vertices, doesn't have a triangle and the minimum degree is greater than $\frac{2n}{5}$, then $G$ is bipartite.

Let $G$ be a triangle-free simple graph whose minimum degree is $> 2n/5$. Assume that $G$ is not a $5$-cycle. Prove that $G$ is bipartite. darij grinberg's note: This is claimed to be a result by Andrásfai, Erdős & Sós (1974) in the Wikipedia,…
Sandeep Ipk
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Maximum number of edges in a bipartite graph

Prove that for a bipartite graph $G$ on $n$ vertices the number of edges in $G$ is at most $\frac{n^2}{4}$. I used induction on $n$. Induction hypothesis: Suppose for a bipartite graph with less than $n$ vertices the result holds true. Now…
sam_rox
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Spanning forests of bipartite graphs and distinct row/column sums of binary matrices

Let $F_{m,n}$ be the set of spanning forests on the complete bipartite graph $K_{m,n}$. Let $$S_{m,n} = \{(r(M), c(M)), M \in B_{m,n} \}$$ where $B_{m,n}$ is the set of $m \times n$ binary matrices and $r(M), c(M)$ are the vectors of row sums and…
Jair Taylor
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Can a bipartite graph have many Hamiltonian paths but no Hamiltonian cycle?

Can a bipartite graph with at least three vertices have the following properties simultaneously: Every vertex is the initial vertex of some Hamiltonian path. The graph contains no Hamiltonian cycle. I found out that this is only possible if the…
Peter
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Prove that all trees are bipartite

I've been trying to prove this for a while. I can think about it intuitively, but I can't come up with a formal proof. I would appreciate some help. Here's how I'm thinking about it Let T be the given tree. Let V(T) be the set of vertices of T. We…
The_Questioner
  • 1,206
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Graph partition that span a third of edges

Given a graph G is easy to see that we have a partition $V=V_1 \cup V_2$ so that $$e(G[V_1])+e(G[V_2])\leq e(G)/2$$. How can we improve this result showing that we can choose $V_i$ such that $e(G[V_i])\leq e(G)/3$ for $i=1,2.$
ghreis
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(Hall's Theorem) Existence of two subfamilies of sets containing the same elements

I came across the following claim in a textbook on combinatorics [1]. Claim (Lindström, Tverberg): Let $A_1, . . . , A_m \subseteq [n]$ be non-empty with $m > n$. There are non-empty, disjoint $I, J \subseteq [m]$, such that $$\bigcup_{i \in I} A_i…
xyz
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Derive Hall's theorem from Tutte's theorem

I'm trying to derive: Hall Theorem A bipartite graph G with partition (A,B) has a matching of A $\Leftrightarrow \forall S\subseteq A, |N(S)|\geq |S|$ From this: Tutte Theorem A graph G has a 1-factor $\Leftrightarrow \forall S\subseteq V(G),…
Marco
  • 111
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The probability of having a perfect matching in a bipartite graph

Say we have a bipartite graph $G$ with two sets, $\{x_1,\dotsc,x_n\}$ and $\{y_1,\dotsc,y_n\}$. For each pair $xy$, there is an edge with probability $p$. Then, what is the probability of having a perfect matching in $G$?
David
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Maximal number of intersections within a bipartite graph

Consider $n$ line segments in the Cartesian plane. For $1\leq k\leq n$, the $k$-th line segment is drawn from $(k,0)$ to $(x_k,1)$, where $\{x_1,x_2,...,x_k\}$ is a permutation of $\{1,2,...,n\}$. Define $a(n)$ to be the maximal number of distinct…
Tracy Morgan
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Eigenvalues of a bipartite graph

Let $X$ be a connected graph with maximum eigenvalue $k$. Assume that $-k$ is also an eigenvalue. I wish to prove that $X$ is bipartite. Now if $\vec{x}=(x_1,\cdots ,x_n)$ is the eigenvector for $-k$ then I can show that for the vector $\vec{y}$…
user10575
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1 answer

Bipartite graphs from permutations

Given are $n\geq 1$ permutations of $abcd$. We construct a bipartite graph $G_{a,b}$ as follows: The $n$ vertices on one side are labeled with the sets containing $a$ and the letters after it in each permutation, and the $n$ vertices on the other…
pi66
  • 7,444
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Colouring bipartite graph with sets of possible colors to each vertex

I'm having some trouble with proving the following: Let $|S(v)|$ be the set of colours available to colour vertex v. The claim is that for every bipartite graph $G=(V,E)$, if $|S(v)| > log_2n$ for each v then it is possible to find a valid colouring…
Gabriel Gava
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