The number of removable edges in a 4-connected graph

AbstractLet G be a 4-connected graph. For an edge e of G, we do the following operations on G: first, delete the edge e from G, resulting the graph G−e; second, for all the vertices x of degree 3 in G−e, delete x from G−e and then completely connect the 3 neighbors of x by a triangle. If multiple edges occur, we use single edges to replace them. The final resultant graph is denoted by G⊖e. If G⊖e is still 4-connected, then e is called a removable edge of G. In this paper we prove that every 4-connected graph of order at least six (excluding the 2-cyclic graph of order six) has at least (4|G|+16)/7 removable edges. We also give the structural characterization of 4-connected graphs for which the lower bound is sharp

On Rainbow Connection Number and Connectivity

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

Basic nets in the projective plane

The notion of basic net (called also basic polyhedron) on $S^2$ plays a central role in Conway's approach to enumeration of knots and links in $S^3$. Drobotukhina applied this approach for links in $RP^3$ using basic nets on $RP^2$. By a result of Nakamoto, all basic nets on $S^2$ can be obtained from a very explicit family of minimal basic nets (the nets $(2\times n)^*$, $n\ge3$, in Conway's notation) by two local transformations. We prove a similar result for basic nets in $RP^2$. We prove also that a graph on $RP^2$ is uniquely determined by its pull-back on $S^3$ (the proof is based on Lefschetz fix point theorem).Comment: 14 pages, 15 figure