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Strongly Monotone Drawings of Planar Graphs
A straight-line drawing of a graph is a monotone drawing if for each pair of
vertices there is a path which is monotonically increasing in some direction,
and it is called a strongly monotone drawing if the direction of monotonicity
is given by the direction of the line segment connecting the two vertices.
We present algorithms to compute crossing-free strongly monotone drawings for
some classes of planar graphs; namely, 3-connected planar graphs, outerplanar
graphs, and 2-trees. The drawings of 3-connected planar graphs are based on
primal-dual circle packings. Our drawings of outerplanar graphs are based on a
new algorithm that constructs strongly monotone drawings of trees which are
also convex. For irreducible trees, these drawings are strictly convex
On two conjectures about the proper connection number of graphs
A path in an edge-colored graph is called proper if no two consecutive edges
of the path receive the same color. For a connected graph , the proper
connection number of is defined as the minimum number of colors
needed to color its edges so that every pair of distinct vertices of are
connected by at least one proper path in . In this paper, we consider two
conjectures on the proper connection number of graphs. The first conjecture
states that if is a noncomplete graph with connectivity and
minimum degree , then , posed by Borozan et al.~in
[Discrete Math. 312(2012), 2550-2560]. We give a family of counterexamples to
disprove this conjecture. However, from a result of Thomassen it follows that
3-edge-connected noncomplete graphs have proper connection number 2. Using this
result, we can prove that if is a 2-connected noncomplete graph with
, then , which solves the second conjecture we want to
mention, posed by Li and Magnant in [Theory \& Appl. Graphs 0(1)(2015), Art.2].Comment: 10 pages. arXiv admin note: text overlap with arXiv:1601.0416
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