Rainbow saturation and graph capacities

The $t$-colored rainbow saturation number $rsat_t(n,F)$ is the minimum size of a $t$-edge-colored graph on $n$ vertices that contains no rainbow copy of $F$, but the addition of any missing edge in any color creates such a rainbow copy. Barrus, Ferrara, Vandenbussche and Wenger conjectured that $rsat_t(n,K_s) = \Theta(n\log n)$ for every $s\ge 3$ and $t\ge \binom{s}{2}$. In this short note we prove the conjecture in a strong sense, asymptotically determining the rainbow saturation number for triangles. Our lower bound is probabilistic in spirit, the upper bound is based on the Shannon capacity of a certain family of cliques.Comment: 5 pages, minor change

The complexity of two graph orientation problems

This is the post-print version of the Article. The official published version can be accessed from the link below - Copyright @ 2012 ElsevierWe consider two orientation problems in a graph, namely the minimization of the sum of all the shortest path lengths and the minimization of the diameter. Our main result is that for each positive integer k, there is a linear-time algorithm that decides for a planar graph Gwhether there is an orientation for which the diameter is at most k. We also extend this result from planar graphs to any minor-closed family F not containing all apex graphs. In contrast, it is known to be NP-complete to decide whether a graph has an orientation such that the sum of all the shortest path lengths is at most an integer specified in the input. We give a simpler proof of this result.This work is partially supported by EC Marie Curie programme NET-ACE (MEST-CT-2004-6724), and Heilbronn Institute for Mathematical Research, Bristol

Separating path systems

We study separating systems of the edges of a graph where each member of the separating system is a path. We conjecture that every $n$-vertex graph admits a separating path system of size $O(n)$ and prove this in certain interesting special cases. In particular, we establish this conjecture for random graphs and graphs with linear minimum degree. We also obtain tight bounds on the size of a minimal separating path system in the case of trees.Comment: 21 pages, fixed misprints, Journal of Combinatoric

Search when the lie depends on the target

The following model is considered. There is exactly one unknown element in the n-element set. A question is a partition of S into three classes: (A,L,B). If x ∈ A then the answer is "yes" (or 1), if x ∈ B then the answer is "no" (or 0), finally if x ∈ L then the answer can be either "yes" or "no". In other words, if the answer "yes" is obtained then we know that x ∈ A ∪ L while in the case of "no" answer the conclusion is x ∈ B ∪ L. The mathematical problem is to minimize the minimum number of questions under certain assumptions on the sizes of A,B and L. This problem has been solved under the condition |L| ≥ k by the author and Krisztián Tichler in previous papers for both the adaptive and non-adaptive cases. In this paper we suggest to solve the problem under the conditions |A| ≤ a, |B| ≤ b. We exhibit some partial results for both the adaptive and non-adaptive cases. We also show that the problem is closely related to some known combinatorial problems. Let us mention that the case b = n - a has been more or less solved in earlier papers. © Springer-Verlag Berlin Heidelberg 2013

Separating Path Systems for the Complete Graph

For any graph $G$, a separating path system of $G$ is a family of paths in $G$ with the property that for any pair of edges in $E(G)$ there is at least one path in the family that contains one edge but not the other. We investigate the size of the smallest separating path system for $K_n$, denoted $f(K_n)$. Our first main result is a construction that shows $f(K_n) \leq \left(\frac{21}{16}+o(1)\right)n$ for sufficiently large $n$. We also show that $f(K_n) \leq n$ whenever $n=p,p+1$ for prime $p$. It is known by simple argument that $f(K_n) \geq n-1$ for all $n \in \mathbb{N}$. A key idea in our construction is to reduce the problem to finding a single path with some particular properties we call a Generator Path. These are defined in such a way that the $n$ cyclic rotations of a generator path provide a separating path system for $K_n$. Hence existence of a generator path for some $K_n$ gives $f(K_n) \leq n$. We construct such paths for all $K_n$ with $n \leq 20$, and show that generator paths exist whenever $n$ is prime.Comment: 23 pages, 3 figure

Some applications of graph theory

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.For the abstract of this thesis, please see the attached PDF.This work was funded by a Marie Curie Early Stage Training Fellowship (NET-ACE-programme) under grant number MEST-CT-2004-6724

Covering and Separation for Permutations and Graphs

This is a thesis of two parts, focusing on covering and separation topics of extremal combinatorics and graph theory, two major themes in this area. They entail the existence and properties of collections of combinatorial objects which together either represent all objects (covering) or can be used to distinguish all objects from each other (separation). We will consider a range of problems which come under these areas. The first part will focus on shattering k-sets with permutations. A family of permutations is said to shatter a given k-set if the permutations cover all possible orderings of the k elements. In particular, we investigate the size of permutation families which cover t orders for every possible k-set as well as study the problem of determining the largest number of k-sets that can be shattered by a family with given size. We provide a construction for a small permutation family which shatters every k-set. We also consider constructions of large families which do not shatter any triple. The second part will be concerned with the problem of separating path systems. A separating path system for a graph is a family of paths where, for any two edges, there is a path containing one edge but not the other. The aim is to find the size of the smallest such family. We will study the size of the smallest separating path system for a range of graphs, including complete graphs, complete bipartite graphs, and lattice-type graphs. A key technique we introduce is the use of generator paths - constructed to utilise the symmetric nature of Kn. We continue this symmetric approach for bipartite graphs and study the limitations of the method. We consider lattice-type graphs as an example of the most efficient possible separating systems for any graph

Separating systems and oriented graphs of diameter two

We prove results on the size of weakly and strongly separating set systems and matrices, and on cross-intersecting systems. As a consequence, we improve on a result of Katona and Szemerédi [6], who proved that the minimal number of edges in an oriented graph of order n with diameter 2 is at least (n/2) log 2(n/2). We show that the minimum is (1 + o(1))n log 2 n.