# Questions tagged [cycles]

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### graphs where every cycle is a sum of triangles

I am studying a special kind of graphs, and I would like to know if they are studied in the literature and what they are called. Let $G$ be a simple, finite, undirected, connected graph, with vertex ...
• 944
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### Construction of graphs of high girth and chromatic number

Are there any concrete constructions of graphs of both high girth and chromatic number? Of course there is the seminal paper of Erdős which proves the existence of such graphs via the probabilistic ...
128 views

### Number of endofunctions in [n] without fixed points with exactly k two-cycles

I need a (numerically) evaluable function for the number $N_{n,k}$ of endofunctions $f: [n] \rightarrow [n]$ without fixed points that have exactly $k$ two-cycles, where $[n] := \{1,\dotsc,n\}$. In ...
• 380
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### Hamming distance globally and Euclidean distance locally to a cycle

Given a permutation matrix 'the question is to decide if there is a permutation matrix representing a cycle within Hamming distance $d$ from given matrix'. Is there an efficient algorithm for it? ...
• 13.1k
1 vote
87 views

### Sequences generated from commuted quaternions and general commuted linear transformations

Given a pair of non-commuting linear transformations, $A$ and $B$, define the "next pair" in a sequence as $A*B$ and $B*A$. I am interested in finite cycles (i.e., the sequence eventually ...
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### Probability permutation in turned to cycle

Let $M$ be a $0/1$ square matrix having one $1$ per row and column (permutation matrix). If you permute the columns and rows independently what is the probability resulting permutation matrix is a ...
• 13.1k
1 vote
359 views

### Expected value of length of longest cycle in permutation

Let $n$ be a positive integer and let $S_n$ be the collection of permutations $\pi:\{1,\ldots,n\}\to \{1,\ldots,n\}$. For $\pi\in S_n$ let $\text{maxcyc}(\pi)$ denote the length of the longest cycle ...
1 vote
216 views

### Unique bipartite perfect matchings and cycles?

Given a graph $G$ which is bipartite and balanced and has unique perfect matching let $G^{e}$ be $G$ without edge $e$. Let $G\cup G_{\pi,\pi'}$ be union of $G$ and $G_{\pi,\pi'}$ where $G_{\pi,\pi'}$ ...
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### A fast algorithm for deciding if a given undirected graph contains a C4 subgraph

I'm looking for an algorithm for deciding if a given undirected graph G contains C4 as a sub graph, not necessarily induced. I'm not interested in finding such a cycle, if it exists. I was told there ...
344 views

### Tree-width of graphs in which any two cycles touch

Let $G$ be a graph s.t. any two cycles $C_1, C_2 \subseteq G$ either have a common vertex or $G$ has an edge joining a vertex in $C_1$ to a vertex of $C_2$. Equivalently: for every cycle $C$ the graph ...
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### Name for specific cycles in graphs

Is there an established name for cycles $C\subseteq G(V,E)$ with the property that $$\lbrace u,v\rbrace\subseteq C\cap V\implies\mathrm{dist}_{|C}(u,v)\le \mathrm{dist}_{|G}(u,v)$$ I would be ...
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### Algorithms for heaviest edge-disjoint cycle collection contained in graph's set of edges

given a biconnected symmetric graph with weighted edges, what is the algorithmic complexity of determining a set of pairwise edge-disjoint cycles with maximal sum of edge weights if there are no other ...
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### Bounding number of k-cycles in a graph

Fix any $k \geq 3$, and suppose I have a simple undirected graph $G=(V,E)$. I want a bound on the number of $k$ cycles in $G$ as a function of $|E|$. In particular, I would like to prove the following ...
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### Strong tournaments

Let $T$ be a strong tournament, and let $N=v_1v_2 \cdots v_n$ be an enumeration of $V(T)$. Let $C$ be a circuit in $T$. We define $i_N(C)=|\{(v_i,v_j) \in E(C); i>j\}|$. Suppose that $N$ is chosen ...
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### A question about dominating circuits in cubic graphs

Let $G$ be a 3-connected cubic graph with a dominating circuit $C$, that is, a circuit such that all edges in $G$ have at least one endvertex in $C$. Let $D$ be another circuit and let the symmetric ...
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### Cycle Structure of a Permutation Based on the Binary Representation

This is a question I posted on math.stackexchange.com before but never got an answer. I am cross-posting it here. Define a permutation $\sigma$ on the set $X=\{1,2,...,n\}$, $n$ is a natural number ...
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1 vote
188 views

### Extremal density of a graph without a non-backtracking $2k$-cycle

The current best bound for the maximum possible density of an $n$-node graph with girth (shortest cycle length) $>2k$ is of the form $$ex(n \ \mid \ C_{\le 2k}) = O(n^{1 + 1/k}),$$ while the ...
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### Cycle class map in smooth quasi-projective varieties

Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$ and $Z$ be a closed subvariety of codimension $k$. Q1. How to define a cycle class $[Z]\in H^k(X,\Omega_X^{k})$ ? Q2. More general, ...
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### Does every connected vertex transitive graph on $n$ vertices (except for $C_n$) have minimum feedback vertex set of size $\Omega(n)$?
Feedback vertex set is a set of vertices whose removal leaves an acyclic graph. It is known that every vertex transitive graph on $n$ vertices has minimum vertex cover of size $\Omega(n)$. It is also ...