Questions tagged [hamiltonian-graphs]
A Hamiltonian graph (directed or undirected) is a graph that contains a Hamiltonian cycle, that is, a cycle that visits every vertex exactly once.
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Infinitely many counterexamples to Nash-Williams's conjecture about hamiltonicity?
Question from 2013
gives one counterexample to Nash-Williams's conjecture about hamiltonicity
of dense digraphs.
Later, we found tens of counterexamples on more than 30 vertices
and believe there are ...
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Two more counterexamples to a conjecture from 1975 about hamiltonicity of digraphs
Question from 2013
gives one counterexample to Nash-Williams's conjecture 1975 about hamiltonicity
of dense digraphs.
In the linked answer, @LouisD "reverse engineered" the counterexample
...
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"Gray code" for building teams
Motivation. In a team of $n$ people, we had the task to build subteams of a fixed size $k<n$ such that every day, $1$ person of the subteam is replaced by another person in the team, but not in the ...
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Is this case of Barnette's Conjecture known?
Context: Barnette's Conjecture is that every bipartite cubic polyhedral graph is Hamiltonian. I have been interested by this problem for a long time, and I recently came up with a result. From my ...
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Is Hamiltonian cycle fixed parameter tractable with parameter clique cover?
Let $G$ be connected simple graph.
Clique cover of graph $G$ is partition of the vertices of $G$
into $k$ disjoint cliques $D'_i$.
Given $G$ and $k$-clique cover, can we solve Hamiltonian cycle
in ...
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Does 2-connectivity imply Hamiltoniancy for subgraphs of the rook graph
We say the rook graph, $R_n$, is the cartesian product of $K_n \times K_n$. Let $S$ be the set of graphs that are an induced subgraph of $R_n$ for some $n$.
Does there exist some constant $c$ such ...
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Extending perfect matchings into Hamiltonian cycles
Let $G$ be a simple cubic graph which has a Hamiltonian circuit $C$. In general, it is not possible to find a second Hamiltonian circuit which contains all the chords of $C$. For example, the Wagner ...
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The number of Hamiltonian circuits on a convex polytope embedded in $\mathbb{R}^N$
Recently I wondered whether there might be a natural topological complexity measure for convex polytopes embedded in $\mathbb{R}^N$. After some reflection it occurred to me that the number of distinct ...
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Normal colorings of bridgeless cubic graphs
Definition (informal) A normal edge-5-coloring of a bridgeless cubic graph $G$ is a proper 5 coloring of the edges of the graph, so that for each edge $e\in E(G)$, either $e$ and the four edges ...
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Arranging all permutations on $\{1,\ldots,n\}$ such that there are no common points
If $n>0$ is an integer, let $[n]=\{1,\ldots,n\}$. Let $S_n$ denote the set of all permutations (bijections) $\pi:[n]\to[n]$.
For which positive integers $n$ is there a bijection $\Phi:[n!]\to S_n$ ...
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Edge colorability and Hamiltonicity of certain classes of cubic graphs (MO graphs)
Let $G$ be a simple cubic graph (that is, 3-regular). A dominating circuit of $G$ is a circuit $C$ such that each edge of $G$ has an endvertex in $C$. The circuit $C$ is chordless if no edge which is ...
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Which are good algorithms for finding Hamiltonian path (not necessarily a circle) up to now?
I am not expertise in graph theory. So have to ask this question here. The term "good" means that the algorithms should be efficient for general undirected simple connected graphs with a higher ...
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Are all cubic graphs almost Hamiltonian?
Call a graph $G$ $n$-almost-Hamiltonian if there is a closed walk in $G$ that visits every vertex of $G$ exactly $n$-times. So a Hamiltonian graph is $n$-almost-Hamiltonian for all $n$. Are all ...
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Is there a permutation $\pi\in S_n$ with $\sum\limits_{0<k<n}\frac1{\pi(k)^2-\pi(k+1)^2}=0$ for each $n>7$?
Let $S_n$ be the symmetric group of all permutations of $\{1,\ldots,n\}$.
QUESTION: Is it true that for each $n=8,9,\ldots$ we have
$$\sum_{0<k<n}\frac1{\pi(k)^2-\pi(k+1)^2}=0\tag{$*$}$$
for ...
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Quantitatively characterizing the failure of the converse of Dirac's theorem
First, I am an undergraduate so I apologize if this is trivial and certainly understand if it is closed immediately.
I am currently in a combinatorics and graph theory class and recently we have ...
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What is the complexity of counting Hamiltonian cycles of a graph?
Since deciding whether a graph contains a Hamiltonian cycle is $NP$-complete, the counting problem which counts the number of such cycles of a graph is $NP$-hard.
Is it also $PP$-hard in the sense ...
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Grinberg's uniquely hamiltonian 3-connected graphs (Russian paper)
Many years ago, Grinberg found some uniquely-hamiltonian $3$-connected graphs, and published his results in a paper that has been cited several times as follows.
E. Grinberg, Three-connected graphs ...
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Halin Graphs with Highest Number of Hamilton Cycles
Halin graphs contain a Hamilton cycle and have the interesting property, that, also in the case of arbitrary real edge weights, it is possible to report one of the shortest contained Hamilton cycles ...
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Reference request: Bipartite symmetric graphs are hamiltonian
Does anyone know whether bipartite symmetric graphs are hamiltonian?
I'm not sure whether anyone have proved it before, but a nonhamiltonian symmetric bipartite graph would lead to a counterexample to ...
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Why is the number of Hamiltonian Cycles of n-octahedron equivalent to the number of Perfect Matching in specific family of Graphs?
In OEIS A003436, it is written that the number of inequivalent labeled Hamilton Cycles of an n-dimesnional Octahedron is the same as the number of Perfect Matchings in a the complement of the Cycle ...
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Are bipartite Moore graphs Hamiltonian?
This is motivated by a computer-generated conjecture that bipartite distance-regular graphs are hamiltonian. I decided to check the case of Moore graphs first.
The cycles and complete bipartite graphs ...
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Maximal non-hamiltonian graphs - spanned by a theta graph?
At the moment I am interested in maximal non-hamiltonian graphs, so that is a (simple, undirected) graph that does not itself have a hamilton cycle, but if you add an edge between any two distinct non-...
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Hamiltonian cycle polytope for the hypercube graph
Let $Q_n$ denote the $n$ dimensional hypercube graph (i.e., graph formed from the vertices and edges of an n-dimensional hypercube). Denote the set of edges and vertices of $Q_n$ by $E_n$ and $V_n$ ...
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Properties of graphs with Hankel-like adjacency matrix
I am having undirected graphs with adjacency matrices which have a regular Hankel-like form, e.g.,
$$A=\begin{pmatrix}0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & (6\times 0 \text{ ...
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Hamiltonicity and minimal degree in bipartite graphs
Given an integer $k>1$, is there a connected bipartite graph $\Gamma = (A, B, E)$ where $A\cap B = \emptyset$ and $E \subseteq \big\{\{a, b\}:a\in A, b\in B\big\}$ such that
$|A| = |B|$,
$\text{...
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Hamiltonian paths in bipartite graphs with 2 sets of "almost" same cardinality
Suppose we have two finite disjoint sets $A, B \neq \emptyset$ such that $|A|$ and $|B|$ differ by at most $1$, and let $\Gamma = (A\cup B, E)$ where $E\subseteq \big\{\{a,b\}: a\in A, b\in B\big\}$ ...
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"Gray code" of all permutations
Informally asking, can we step through all permutations of the set $\{1,\ldots,n\}$ by just using transpositions?
More formally: For any $n\in\mathbb{N}$ let $[n] = \{1,\ldots,n\}$ and let $S_n$ be ...
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What was Smith's proof of Smith's theorem on Hamilton cycles in cubic graphs?
In a short 1946 paper "On Hamiltonian Circuits", Tutte proved the famous result that an edge in a cubic graph lies in an even number of Hamilton circuits.
He attributed the result to his friend CAB ...
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Does every finite bridgeless cubic planar simple undirected graph admit a 2-factorization with at most two components each of which has even order?
Consider simple bridgeless cubic planar graphs.
Does each such graph admit a 2-factorization with $\leq 2$ components each of which has even order?
If not, does anyone know of an counterexample?
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Number Associated with Straight-line Drawings of Hamiltonian Graphs
Is there anything known about the maximum number of simple-polygonal Hamilton cycles that a straight-line drawing of a Hamiltonian graph can have?
Put differently, if the vertices of a Hamilton ...
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Counting cycle vertex covers on hypercube
Let $Q_n$ be the $n$-dimensional hypercube graph. How many vertex cycle covers exist on $Q_n$? (Presumably the best we can hope for are upper and lower bounds.) To be clear, a single "vertex cycle ...
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Heuristic arguments regarding Sheehan's conjecture?
Sheehan conjectured that there are no 4-regular graphs that are uniquely hamiltonian (i.e. have exactly one hamilton cycle).
Evidence that might be loosely seen to be in favour of this conjecture is: ...
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Is there a non-bipartite hamiltonian cubic graph on $n$ vertices with no $(n-1)$-cycle?
Is there a cubic (3-regular) graph $G$ on $n$ vertices such that:
$G$ is hamiltonian
$G$ has no $(n-1)$-cycles
$G$ is not bipartite
My computer tells me that there are none on up to $24$ vertices.
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Minimum number of hamilton cycles in a 4-connected planar triangulation?
I am currently interested in hamilton cycles (i.e. a cycle through every vertex) in planar triangulations (i.e. planar graphs with every face a triangle).
There are non-hamiltonian planar ...
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Graph gadget related to uniquely hamiltionian regular graphs (question #2)
Related to uniquely hamiltionian graphs.
For natural numbers $a,b$ define $(a,b)$ gadget $G$:
$G$ is finite simple graph. Two vertices $u,v$ are of degree $b$
and the rest of the vertices are of ...
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Graph gadget related to uniquely hamiltionian regular graphs
A graph is uniquely hamiltonian if it has exactly one hamiltonian cycle.
According to a conjecture there are no $r$-regular uniquely hamiltonian
graphs for $r > 2$ and of special interest is the ...
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Cage graphs and even cycles
Let $G$ be a $(\nu,g)$-cage graph of degree $\nu$ with girth $g$ and $n=n(\nu,g)$ vertices.
Based on the known examples, I am wondering if the following can be proved/disproved:
Is it true that ...
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What is the smallest uniquely hamiltonian graph with minimum degree at least 3?
I would like to know more about uniquely hamiltonian graphs with minimum vertex degree at least 3, and in particular what is the smallest one.
(Recall that a graph is hamiltonian if it has a cycle ...
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Localizing Bondy's metaconjecture on hamiltonicity
Definitions:
Let $G$ be a graph on $n$ vertices. $G$ is Hamiltonian provided $G$ has a cycle of length $n$. $G$ is pancyclic provided $G$ has a cycle of length $\ell$ for every $3 \leq \ell \leq n$.
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degG(x) + degG(y) >= n, show that the graph is hamiltonian [closed]
I'm trying to show that a connected graph which has order >=3, and having the following inequality is Hamiltonian:
...
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How many edges can be added to two circles before the graph becomes Hamiltonian?
Start with two $n$-circles $(v_1\cdots v_n)$ and $(w_1\cdots w_n)$ of vertice sets $V$ and $W$, where $n\ge 5$. Add a number of vertex-disjoint edges between $V$ and $W$ (thus no chords) in a way ...
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What is the densest bipartite graph with unique Hamiltonian cycle?
In a prior post regarding perfect matching, it was stated that the densest graph with a unique perfect matching cannot have more than $n^2$ edges, if graph has $2n$ vertices.
Analogously, what is the ...
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What is the complexity of finding a third Hamilton Cycle in cubic graph?
According to Smith Theorem: if a cubic graph has a hamilton circuit then it must have a second one. SMITH : Given a Hamilton circuit in a 3-regular graph, find a second Hamilton circuit. It is known ...
user39815
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Simple decomposition of $K_{2n}-I$ into hamiltonian cycles
http://mathworld.wolfram.com/HamiltonDecomposition.html
In the 1890s, Walecki showed that complete graphs K_n admit a Hamilton decomposition for odd n, and decompositions into Hamiltonian cycles ...
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Refinement of Dirac's theorem on Hamiltonian graphs
Dirac's theorem states that if degree of each vertex of a graph $G=(V,E)$ is not less than $|V|/2$, then it has Hamiltonian cycle. It is less known, but still known and not so hard to prove (though I ...
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Can we find 3 disjoint directed Hamiltonian cycles in the cube?
Let $D$ be the digraph on $2^d$ vertices with $d2^d$ edges that we obtain by directing each edge of the $d$-dimensional hypercube in both directions.
Can we partition the edges of $D$ into $d$ ...
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A digraph related to permutations
A finite sequence of distinct real numbers of length $n$ determines a linear order of $\{1,\ldots,n\}$, by mapping position to rank; call this the permutation of the sequence.
Consider the following ...
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How many hamiltonian cycles can be removed from a complete directed graph before it becomes disconnected?
The question started from a problem brought home by a friend's 5th grader: "How many ways can you seat 5 people around a round table so that the people sitting to the left of any person is different ...
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Hamiltonicity of random graphs with high girth
We say that $G\sim G_{n,f}$ (for $f=f(n)$) if $G$ is chosen uniformly at random from all graphs on $n$ vertices with girth $g(G)\ge f(n)$. Is there any threshold function $F(n)$ such that when $f\ll F$...
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Knight's tours in higher dimensions
I wonder if Knight's Tours have been explored in higher dimensions,
using the following definition of a knight move.
In dimension $d=2$, the knight moves left/right and forward/back
one step and two ...
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