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        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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        36 views

        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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        389 views

        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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        1answer
        99 views

        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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        1answer
        109 views

        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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        1answer
        736 views

        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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        131 views

        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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        155 views

        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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        1answer
        129 views

        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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        1answer
        150 views

        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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        108 views

        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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        69 views

        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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        1answer
        105 views

        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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        1answer
        318 views

        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 ...
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        36 views

        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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        54 views

        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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