69,267 research outputs found
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
On the oriented chromatic number of dense graphs
Let be a graph with vertices, edges, average degree , and maximum degree . The \emph{oriented chromatic number} of is the maximum, taken over all orientations of , of the minimum number of colours in a proper vertex colouring such that between every pair of colour classes all edges have the same orientation. We investigate the oriented chromatic number of graphs, such as the hypercube, for which . We prove that every such graph has oriented chromatic number at least . In the case that , this lower bound is improved to . Through a simple connection with harmonious colourings, we prove a general upper bound of \Oh{\Delta\sqrt{n}} on the oriented chromatic number. Moreover this bound is best possible for certain graphs. These lower and upper bounds are particularly close when is ()-regular for some constant , in which case the oriented chromatic number is between and
On Rainbow Connection Number and Connectivity
Rainbow connection number, , of a connected graph is the minimum
number of colours needed to colour its edges, so that every pair of vertices is
connected by at least one path in which no two edges are coloured the same. In
this paper we investigate the relationship of rainbow connection number with
vertex and edge connectivity. It is already known that for a connected graph
with minimum degree , the rainbow connection number is upper bounded by
[Chandran et al., 2010]. This directly gives an upper
bound of and for rainbow
connection number where and , respectively, denote the edge
and vertex connectivity of the graph. We show that the above bound in terms of
edge connectivity is tight up-to additive constants and show that the bound in
terms of vertex connectivity can be improved to , for any . We conjecture that rainbow connection
number is upper bounded by and show that it is true for
. We also show that the conjecture is true for chordal graphs and
graphs of girth at least 7.Comment: 10 page
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