169 research outputs found
Independent Sets in Graphs with an Excluded Clique Minor
Let be a graph with vertices, with independence number , and
with with no -minor for some . It is proved that
Cyclic Coloring of Plane Graphs with Maximum Face Size 16 and 17
Plummer and Toft conjectured in 1987 that the vertices of every 3-connected
plane graph with maximum face size D can be colored using at most D+2 colors in
such a way that no face is incident with two vertices of the same color. The
conjecture has been proven for D=3, D=4 and D>=18. We prove the conjecture for
D=16 and D=17
Hadwiger's conjecture for graphs with forbidden holes
Given a graph , the Hadwiger number of , denoted by , is the
largest integer such that contains the complete graph as a minor.
A hole in is an induced cycle of length at least four. Hadwiger's
Conjecture from 1943 states that for every graph , , where
denotes the chromatic number of . In this paper we establish more
evidence for Hadwiger's conjecture by showing that if a graph with
independence number has no hole of length between and
, then . We also prove that if a graph with
independence number has no hole of length between and
, then contains an odd clique minor of size , that is,
such a graph satisfies the odd Hadwiger's conjecture
Third case of the Cyclic Coloring Conjecture
The Cyclic Coloring Conjecture asserts that the vertices of every plane graph
with maximum face size D can be colored using at most 3D/2 colors in such a way
that no face is incident with two vertices of the same color. The Cyclic
Coloring Conjecture has been proven only for two values of D: the case D=3 is
equivalent to the Four Color Theorem and the case D=4 is equivalent to
Borodin's Six Color Theorem, which says that every graph that can be drawn in
the plane with each edge crossed by at most one other edge is 6-colorable. We
prove the case D=6 of the conjecture
On a special case of Hadwiger's conjecture
Hadwiger's Conjecture seems difficult to attack, even in the very special case of graphs G of independence number α (G) = 2. We present some results in this special case
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