We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, $t$-tough graphs, and claw-free graphs

Broersma, H.J.

University of Twente Research Information

English

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Rainbow connection number, $rc(G)$, of a connected graph $G$ 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 $\delta$, the rainbow connection number is upper bounded by
$3n/(\delta + 1) + 3$ [Chandran et al., 2010]. This directly gives an upper
bound of $3n/(\lambda + 1) + 3$ and $3n/(\kappa + 1) + 3$ for rainbow
connection number where $\lambda$ and $\kappa$, 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 $(2 + \epsilon)n/\kappa + 23/
\epsilon^2$, for any $\epsilon > 0$. We conjecture that rainbow connection
number is upper bounded by $n/\kappa + O(1)$ and show that it is true for
$\kappa = 2$. We also show that the conjecture is true for chordal graphs and
graphs of girth at least 7.Comment: 10 page