![](/files/happy5.png)
Cycles
Cycle double cover conjecture ★★★★
(m,n)-cycle covers ★★★
Author(s): Celmins; Preissmann
Faithful cycle covers ★★★
Author(s): Seymour
![$ G = (V,E) $](/files/tex/5969f28fd067291799f25ca43b6642feb6b04bd0.png)
![$ p : E \rightarrow {\mathbb Z} $](/files/tex/acf577dca5adcf9fd9f7fb631a68262035044887.png)
![$ p(e) $](/files/tex/fa56cd603dd6dbffa93ed375e6a002107e59c9bb.png)
![$ e \in E(G) $](/files/tex/730c5d64c8d749c640adc18eb493c641ff1addc9.png)
![$ (G,p) $](/files/tex/d9c8f5e55f04622be55791c713068d286259ce27.png)
Decomposing eulerian graphs ★★★
Author(s):
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ P $](/files/tex/b2b0b759db4d5a1b3204c38cdee6d9bd9e0d0dab.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ (G,P) $](/files/tex/fd3ab98e5fb37dd2a932a24c976ad96933b74a0e.png)
Barnette's Conjecture ★★★
Author(s): Barnette
Keywords: bipartite; cubic; hamiltonian
r-regular graphs are not uniquely hamiltonian. ★★★
Author(s): Sheehan
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ r $](/files/tex/535dee6c3b72bcc4d571239ed00be162ee1e6fbe.png)
![$ r > 2 $](/files/tex/97a6c1471f5d46c5d21bf3e382139df59f21bd80.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
Keywords: hamiltonian; regular; uniquely hamiltonian
Hamiltonian cycles in line graphs ★★★
Author(s): Thomassen
Keywords: hamiltonian; line graphs
Geodesic cycles and Tutte's Theorem ★★
Author(s): Georgakopoulos; Sprüssel
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ 3 $](/files/tex/4aaf85facb6534fd470edd32dbdb4e28f6218190.png)
![$ \ell: E(G) \to \mathb R^+ $](/files/tex/aad2d29bf9a23e904e02b7e5f616c403cb1b95df.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ \ell $](/files/tex/d2c5960dd9795a1b000a5843d282c97268e303c4.png)
Keywords: cycle space; geodesic cycles; peripheral cycles
Jones' conjecture ★★
For a graph , let
denote the cardinality of a maximum cycle packing (collection of vertex disjoint cycles) and let
denote the cardinality of a minimum feedback vertex set (set of vertices
so that
is acyclic).
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ cc(G)\leq 2cp(G) $](/files/tex/c3407aab8c5dd67b27ec62b418dcd71d7b6fb886.png)
Keywords: cycle packing; feedback vertex set; planar graph
Chords of longest cycles ★★★
Author(s): Thomassen
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
Keywords: chord; connectivity; cycle
Hamiltonicity of Cayley graphs ★★★
Author(s): Rapaport-Strasser
Keywords:
Strong 5-cycle double cover conjecture ★★★
Author(s): Arthur; Hoffmann-Ostenhof
![$ C $](/files/tex/05d3558ef52267cc1af40e658352d98883668ee3.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ C $](/files/tex/05d3558ef52267cc1af40e658352d98883668ee3.png)
Keywords: cycle cover
Decomposing an eulerian graph into cycles. ★★
Author(s): Hajós
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ \frac{1}{2}(n-1) $](/files/tex/917a680b31340814d24e8e9ea8f3fe6219d868ad.png)
Keywords:
Decomposing an eulerian graph into cycles with no two consecutives edges on a prescribed eulerian tour. ★★
Author(s): Sabidussi
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ 4 $](/files/tex/1f1498726bb4b7754ca36de46c0ccdd09136d115.png)
![$ W $](/files/tex/48948390b5dc5ab9a52c0afaff3e950050be14a2.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ W $](/files/tex/48948390b5dc5ab9a52c0afaff3e950050be14a2.png)
Keywords:
Every prism over a 3-connected planar graph is hamiltonian. ★★
Author(s): Kaiser; Král; Rosenfeld; Ryjácek; Voss
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ 3 $](/files/tex/4aaf85facb6534fd470edd32dbdb4e28f6218190.png)
![$ G\square K_2 $](/files/tex/eb7d7828977b0484fd118a68143d72f9c6e865f3.png)
Keywords:
4-connected graphs are not uniquely hamiltonian ★★
Author(s): Fleischner
![$ 4 $](/files/tex/1f1498726bb4b7754ca36de46c0ccdd09136d115.png)
Keywords:
Hamilton decomposition of prisms over 3-connected cubic planar graphs ★★
![$ 3 $](/files/tex/4aaf85facb6534fd470edd32dbdb4e28f6218190.png)
Keywords:
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