127 research outputs found
Multiple Petersen subdivisions in permutation graphs
A permutation graph is a cubic graph admitting a 1-factor M whose complement
consists of two chordless cycles. Extending results of Ellingham and of
Goldwasser and Zhang, we prove that if e is an edge of M such that every
4-cycle containing an edge of M contains e, then e is contained in a
subdivision of the Petersen graph of a special type. In particular, if the
graph is cyclically 5-edge-connected, then every edge of M is contained in such
a subdivision. Our proof is based on a characterization of cographs in terms of
twin vertices. We infer a linear lower bound on the number of Petersen
subdivisions in a permutation graph with no 4-cycles, and give a construction
showing that this lower bound is tight up to a constant factor
Two floor building needing eight colors
Motivated by frequency assignment in office blocks, we study the chromatic
number of the adjacency graph of -dimensional parallelepiped arrangements.
In the case each parallelepiped is within one floor, a direct application of
the Four-Colour Theorem yields that the adjacency graph has chromatic number at
most . We provide an example of such an arrangement needing exactly
colours. We also discuss bounds on the chromatic number of the adjacency graph
of general arrangements of -dimensional parallelepipeds according to
geometrical measures of the parallelepipeds (side length, total surface or
volume)
Fractional coloring of triangle-free planar graphs
We prove that every planar triangle-free graph on vertices has fractional
chromatic number at most
Every plane graph of maximum degree 8 has an edge-face 9-colouring
An edge-face colouring of a plane graph with edge set and face set is
a colouring of the elements of such that adjacent or incident
elements receive different colours. Borodin proved that every plane graph of
maximum degree can be edge-face coloured with colours.
Borodin's bound was recently extended to the case where . In this
paper, we extend it to the case .Comment: 29 pages, 1 figure; v2 corrects a contraction error in v1; to appear
in SIDM
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