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DeVos, Matt


Graph Theory » Basic G.T.

Friendly partitions ★★

Author(s): DeVos

A friendly partition of a graph is a partition of the vertices into two sets so that every vertex has at least as many neighbours in its own class as in the other.

Problem   Is it true that for every $ r $, all but finitely many $ r $-regular graphs have friendly partitions?

Keywords: edge-cut; partition; regular

Posted by mdevos
updated August 5th, 2020
1 comment
Graph Theory » Coloring » Vertex coloring

Circular colouring the orthogonality graph ★★

Author(s): DeVos; Ghebleh; Goddyn; Mohar; Naserasr

Let $ {\mathcal O} $ denote the graph with vertex set consisting of all lines through the origin in $ {\mathbb R}^3 $ and two vertices adjacent in $ {\mathcal O} $ if they are perpendicular.

Problem   Is $ \chi_c({\mathcal O}) = 4 $?

Keywords: circular coloring; geometric graph; orthogonality

Posted by mdevos
updated September 28th, 2009
4 comments
Graph Theory » Topological G.T. » Coloring

5-local-tensions ★★

Author(s): DeVos

Conjecture   There exists a fixed constant $ c $ (probably $ c=4 $ suffices) so that every embedded (loopless) graph with edge-width $ \ge c $ has a 5-local-tension.

Keywords: coloring; surface; tension

Posted by mdevos
updated June 22nd, 2007
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Number Theory » Combinatorial N.T.

Gao's theorem for nonabelian groups ★★

Author(s): DeVos

For every finite multiplicative group $ G $, let $ s(G) $ ($ s'(G) $) denote the smallest integer $ m $ so that every sequence of $ m $ elements of $ G $ has a subsequence of length $ >0 $ (length $ |G| $) which has product equal to 1 in some order.

Conjecture   $ s'(G) = s(G) + |G| - 1 $ for every finite group $ G $.

Keywords: subsequence sum; zero sum

Posted by mdevos
updated May 23rd, 2007
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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 » Topological G.T. » Crossing numbers

Drawing disconnected graphs on surfaces ★★

Author(s): DeVos; Mohar; Samal

Conjecture   Let $ G $ be the disjoint union of the graphs $ G_1 $ and $ G_2 $ and let $ \Sigma $ be a surface. Is it true that every optimal drawing of $ G $ on $ \Sigma $ has the property that $ G_1 $ and $ G_2 $ are disjoint?

Keywords: crossing number; surface

Posted by mdevos
updated September 30th, 2007
1 comment
Graph Theory » Topological G.T. » Planar graphs

What is the largest graph of positive curvature? ★

Author(s): DeVos; Mohar

Problem   What is the largest connected planar graph of minimum degree 3 which has everywhere positive combinatorial curvature, but is not a prism or antiprism?

Keywords: curvature; planar graph

Posted by mdevos
updated August 30th, 2011
2 comments
Graph Theory » Basic G.T. » Connectivity

Partitioning edge-connectivity ★★

Author(s): DeVos

Question   Let $ G $ be an $ (a+b+2) $-edge-connected graph. Does there exist a partition $ \{A,B\} $ of $ E(G) $ so that $ (V,A) $ is $ a $-edge-connected and $ (V,B) $ is $ b $-edge-connected?

Keywords: edge-coloring; edge-connectivity

Posted by mdevos
updated March 7th, 2007
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Graph Theory » Coloring » Edge coloring

Packing T-joins ★★

Author(s): DeVos

Conjecture   There exists a fixed constant $ c $ (probably $ c=1 $ suffices) so that every graft with minimum $ T $-cut size at least $ k $ contains a $ T $-join packing of size at least $ (2/3)k-c $.

Keywords: packing; T-join

Posted by mdevos
updated March 7th, 2007
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Graph Theory » Coloring » Nowhere-zero flows

A homomorphism problem for flows ★★

Author(s): DeVos

Conjecture   Let $ M,M' $ be abelian groups and let $ B \subseteq M $ and $ B' \subseteq M' $ satisfy $ B=-B $ and $ B' = -B' $. If there is a homomorphism from $ Cayley(M,B) $ to $ Cayley(M',B') $, then every graph with a B-flow has a B'-flow.

Keywords: homomorphism; nowhere-zero flow; tension

Posted by mdevos
updated March 7th, 2007
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