![](/files/happy5.png)
Seymour, Paul D.
Fractional Hadwiger ★★
Author(s): Harvey; Reed; Seymour; Wood
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
(a)
![$ \chi_f(G)\leq\text{had}(G) $](/files/tex/50fb973c4dd31a8fde6ae9c6a9ba74c3eca2849a.png)
(b)
![$ \chi(G)\leq\text{had}_f(G) $](/files/tex/6ef2849ce5271ed2eb983602db9e4948dcc30e87.png)
(c)
![$ \chi_f(G)\leq\text{had}_f(G) $](/files/tex/7e131f28dcdff73d40ea8cfaf990a2ad70fb9952.png)
Keywords: fractional coloring, minors
Seymour's r-graph conjecture ★★★
Author(s): Seymour
An -graph is an
-regular graph
with the property that
for every
with odd size.
![$ \chi'(G) \le r+1 $](/files/tex/efa38d9230a3451d1c38c061522cb607572d369b.png)
![$ r $](/files/tex/535dee6c3b72bcc4d571239ed00be162ee1e6fbe.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
Keywords: edge-coloring; r-graph
Non-edges vs. feedback edge sets in digraphs ★★★
Author(s): Chudnovsky; Seymour; Sullivan
For any simple digraph , we let
be the number of unordered pairs of nonadjacent vertices (i.e. the number of non-edges), and
be the size of the smallest feedback edge set.
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ \le 3 $](/files/tex/2dbc994c29a8272f2fa20bee6216f47315c47aa7.png)
![$ \beta(G) \le \frac{1}{2} \gamma(G) $](/files/tex/d2838535a0339a448fba3bbbab8586020be5f886.png)
Keywords: acyclic; digraph; feedback edge set; triangle free
Seagull problem ★★★
Author(s): Seymour
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ 3 $](/files/tex/4aaf85facb6534fd470edd32dbdb4e28f6218190.png)
![$ \ge \frac{n}{2} $](/files/tex/4efffa1e5aca5aa0354077be96157068fab5f8be.png)
Keywords: coloring; complete graph; minor
Seymour's Second Neighbourhood Conjecture ★★★
Author(s): Seymour
Keywords: Caccetta-Häggkvist; neighbourhood; second; Seymour
Bases of many weights ★★★
Let be an (additive) abelian group, and for every
let
.
![$ M $](/files/tex/3f02401f624e31ef8679d3c3628c1f310058f388.png)
![$ E $](/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png)
![$ w : E \rightarrow G $](/files/tex/7c1a9b6ba2a67b002e25339803f3d5a6da1b684d.png)
![$ S = \{ \sum_{b \in B} w(b) : B \mbox{ is a base} \} $](/files/tex/15b6767566360480b09b1982f902c978265de9c9.png)
![$ H = {\mathit stab}(S) $](/files/tex/e338cf1ff9295e47c3f1552771eb4fecfb4d730f.png)
![$$|S| \ge |H| \left( 1 - rk(M) + \sum_{Q \in G/H} rk(w^{-1}(Q)) \right).$$](/files/tex/e22b2f4b60bf6bbfd304dd219b3092fb50cbae67.png)
Alon-Saks-Seymour Conjecture ★★★
Author(s): Alon; Saks; Seymour
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ m $](/files/tex/ddaab6dc091926fb1da549195000491cefae85c1.png)
![$ \chi(G) \le m+1 $](/files/tex/0aefe0f752ac272776eec5c61c7d1a1822a3d224.png)
Keywords: coloring; complete bipartite graph; eigenvalues; interlacing
Seymour's self-minor conjecture ★★★
Author(s): Seymour
Keywords: infinite graph; minor
Faithful cycle covers ★★★
Author(s): Seymour
![$ 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)
Cycle double cover conjecture ★★★★
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