Combinatorics » Matroid Theory

Equality in a matroidal circumference bound ★★

Author(s): Oxley; Royle

Question   Is the binary affine cube $ AG(3,2) $ the only 3-connected matroid for which equality holds in the bound $$E(M) \leq c(M) c(M^*) / 2$$ where $ c(M) $ is the circumference (i.e. largest circuit size) of $ M $?

Keywords: circumference

Posted by Gordon Royle
updated November 2nd, 2007
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Graph Theory » Infinite Graphs

Highly arc transitive two ended digraphs ★★

Author(s): Cameron; Praeger; Wormald

Conjecture   If $ G $ is a highly arc transitive digraph with two ends, then every tile of $ G $ is a disjoint union of complete bipartite graphs.

Keywords: arc transitive; digraph; infinite graph

Posted by mdevos
updated October 29th, 2007
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Graph Theory » Infinite Graphs

Strong matchings and covers ★★★

Author(s): Aharoni

Let $ H $ be a hypergraph. A strongly maximal matching is a matching $ F \subseteq E(H) $ so that $ |F' \setminus F| \le |F \setminus F'| $ for every matching $ F' $. A strongly minimal cover is a (vertex) cover $ X \subseteq V(H) $ so that $ |X' \setminus X| \ge |X \setminus X'| $ for every cover $ X' $.

Conjecture   If $ H $ is a (possibly infinite) hypergraph in which all edges have size $ \le k $ for some integer $ k $, then $ H $ has a strongly maximal matching and a strongly minimal cover.

Keywords: cover; infinite graph; matching

Posted by mdevos
updated October 23rd, 2007
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Graph Theory » Infinite Graphs

Unfriendly partitions ★★★

Author(s): Cowan; Emerson

If $ G $ is a graph, we say that a partition of $ V(G) $ is unfriendly if every vertex has at least as many neighbors in the other classes as in its own.

Problem   Does every countably infinite graph have an unfriendly partition into two sets?

Keywords: coloring; infinite graph; partition

Posted by mdevos
updated October 22nd, 2007
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