Seymour, Paul D.


Fractional Hadwiger ★★

Author(s): Harvey; Reed; Seymour; Wood

Conjecture   For every graph $ G $,
(a) $ \chi_f(G)\leq\text{had}(G) $
(b) $ \chi(G)\leq\text{had}_f(G) $
(c) $ \chi_f(G)\leq\text{had}_f(G) $.

Keywords: fractional coloring, minors

Seymour's r-graph conjecture ★★★

Author(s): Seymour

An $ r $-graph is an $ r $-regular graph $ G $ with the property that $ |\delta(X)| \ge r $ for every $ X \subseteq V(G) $ with odd size.

Conjecture   $ \chi'(G) \le r+1 $ for every $ r $-graph $ G $.

Keywords: edge-coloring; r-graph

Non-edges vs. feedback edge sets in digraphs ★★★

Author(s): Chudnovsky; Seymour; Sullivan

For any simple digraph $ G $, we let $ \gamma(G) $ be the number of unordered pairs of nonadjacent vertices (i.e. the number of non-edges), and $ \beta(G) $ be the size of the smallest feedback edge set.

Conjecture  If $ G $ is a simple digraph without directed cycles of length $ \le 3 $, then $ \beta(G) \le \frac{1}{2} \gamma(G) $.

Keywords: acyclic; digraph; feedback edge set; triangle free

Seagull problem ★★★

Author(s): Seymour

Conjecture   Every $ n $ vertex graph with no independent set of size $ 3 $ has a complete graph on $ \ge \frac{n}{2} $ vertices as a minor.

Keywords: coloring; complete graph; minor

Seymour's Second Neighbourhood Conjecture ★★★

Author(s): Seymour

Conjecture   Any oriented graph has a vertex whose outdegree is at most its second outdegree.

Keywords: Caccetta-Häggkvist; neighbourhood; second; Seymour

Bases of many weights ★★★

Author(s): Schrijver; Seymour

Let $ G $ be an (additive) abelian group, and for every $ S \subseteq G $ let $ {\mathit stab}(S) = \{ g \in G : g + S = S \} $.

Conjecture   Let $ M $ be a matroid on $ E $, let $ w : E \rightarrow G $ be a map, put $ S = \{ \sum_{b \in B} w(b) : B \mbox{ is a base} \} $ and $ H = {\mathit stab}(S) $. Then $$|S| \ge |H| \left( 1 - rk(M) + \sum_{Q \in G/H} rk(w^{-1}(Q)) \right).$$

Keywords: matroid; sumset; zero sum

Alon-Saks-Seymour Conjecture ★★★

Author(s): Alon; Saks; Seymour

Conjecture   If $ G $ is a simple graph which can be written as an union of $ m $ edge-disjoint complete bipartite graphs, then $ \chi(G) \le m+1 $.

Keywords: coloring; complete bipartite graph; eigenvalues; interlacing

Seymour's self-minor conjecture ★★★

Author(s): Seymour

Conjecture   Every infinite graph is a proper minor of itself.

Keywords: infinite graph; minor

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.

Keywords: cover; cycle

Cycle double cover conjecture ★★★★

Author(s): Seymour; Szekeres

Conjecture   For every graph with no bridge, there is a list of cycles so that every edge is contained in exactly two.

Keywords: cover; cycle

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