cycle

Graph Theory » Basic G.T. » Cycles

Chords of longest cycles ★★★

Author(s): Thomassen

Conjecture If $G$ is a 3-connected graph, every longest cycle in $G$ has a chord.

Keywords: chord; connectivity; cycle

Posted by mdevos
updated October 14th, 2023

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Graph Theory » Extremal G.T.

What is the smallest number of disjoint spanning trees made a graph Hamiltonian ★★

Author(s): Goldengorin

We are given a complete simple undirected weighted graph $G_1=(V,E)$ and its first arbitrary shortest spanning tree $T_1=(V,E_1)$ . We define the next graph $G_2=(V,E\setminus E_1)$ and find on $G_2$ the second arbitrary shortest spanning tree $T_2=(V,E_2)$ . We continue similarly by finding $T_3=(V,E_3)$ on $G_3=(V,E\setminus \cup_{i=1}^{2}E_i)$ , etc. Let k be the smallest number of disjoint shortest spanning trees as defined above and let $T^{k}=(V,\cup_{i=1}^{k}E_i)$ be the graph obtained as union of all $k$ disjoint trees.

Question 1. What is the smallest number of disjoint spanning trees creates a graph $T^{k}$ containing a Hamiltonian path.

Question 2. What is the smallest number of disjoint spanning trees creates a graph $T^{k}$ containing a shortest Hamiltonian path?

Questions 3 and 4. Replace in questions 1 and 2 a shortest spanning tree by a 1-tree. What is the smallest number of disjoint 1-trees creates a Hamiltonian graph? What is the smallest number of disjoint 1-trees creates a graph containing a shortest Hamiltonian cycle?

Keywords: 1-trees; cycle; Hamitonian path; spanning trees

Posted by boris
updated September 10th, 2007

1 comment

Graph Theory » Basic G.T. » Cycles

Bigger cycles in cubic graphs ★★

Author(s):

Problem Let $G$ be a cyclically 4-edge-connected cubic graph and let $C$ be a cycle of $G$ . Must there exist a cycle $C' \neq C$ so that $V(C) \subseteq V(C')$ ?

Keywords: cubic; cycle

Posted by mdevos
updated September 4th, 2009

1 comment

Graph Theory » Coloring » Nowhere-zero flows

Antichains in the cycle continuous order ★★

Author(s): DeVos

If $G$ , $H$ are graphs, a function $f : E(G) \rightarrow E(H)$ is called cycle-continuous if the pre-image of every element of the (binary) cycle space of $H$ is a member of the cycle space of $G$ .

Problem Does there exist an infinite set of graphs $\{G_1,G_2,\ldots \}$ so that there is no cycle continuous mapping between $G_i$ and $G_j$ whenever $i \neq j$ ?

Keywords: antichain; cycle; poset

Posted by mdevos
updated May 12th, 2007

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Graph Theory » Algebraic G.T.

Hamiltonian paths and cycles in vertex transitive graphs ★★★

Author(s): Lovasz

Problem Does every connected vertex-transitive graph have a Hamiltonian path?

Keywords: cycle; hamiltonian; path; vertex-transitive

Posted by mdevos
updated March 18th, 2007

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Graph Theory » Basic G.T. » Cycles

Decomposing eulerian graphs ★★★

Author(s):

Conjecture If $G$ is a 6-edge-connected Eulerian graph and $P$ is a 2-transition system for $G$ , then $(G,P)$ has a compaible decomposition.

Keywords: cover; cycle; Eulerian

Posted by mdevos
updated March 7th, 2007

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Graph Theory » Basic G.T. » Cycles

Faithful cycle covers ★★★

Author(s): Seymour

Conjecture If $G = (V,E)$ is a graph, $p : E \rightarrow {\mathbb Z}$ is admissable, and $p(e)$ is even for every $e \in E(G)$ , then $(G,p)$ has a faithful cover.