arXiv subjects
Publications and source records attributed to Gabor Meszaros.
A graph is path-pairable if for any pairing of its vertices there exist edge disjoint paths joining the vertices in each pair. We obtain sharp bounds on the maximum possible diameter of path-pairable graphs which either have a given number of edges, or are c- degenerate. Along the way we show that a large family of graphs obtained by blowing up a path is path-pairable, which may be of independent interest.
In this paper we investigate the problem of sorting a set of $n$ coins, each with distinct but unknown weights, using an unusual scale. The classical version of this problem, which has been well-studied, gives the user a binary scale, enabling them to determine which is the lighter/heavier of any two objects. We generalise this, considering a scale that accepts $k$ coins as input and returns the $t^{\text{th}}$ lightest, for a fixed $k$ and $t$. We consider this in both an on-line and off-line setting, and exhibit algorithms in both settings that are best-possible in terms of the order of the number of queries required.
A triple of vertices in a graph is a \emph{frustrated triangle} if it induces an odd number of edges. We study the set $F_n\subset[0,\binom{n}{3}]$ of possible number of frustrated triangles $f(G)$ in a graph $G$ on $n$ vertices. We prove that about two thirds of the numbers in $[0,n^{3/2}]$ cannot appear in $F_n$, and we characterise the graphs $G$ with $f(G)\in[0,n^{3/2}]$. More precisely, our main result is that, for each $n\geq 3$, $F_n$ contains two interlacing sequences $0=a_0\leq b_0\leq a_1\leq b_1\leq \dots \leq a_m\leq b_m\sim n^{3/2}$ such that $F_n\cap(b_t,a_{t+1})=\emptyset$ for all $t$, where the gaps are $|b_t-a_{t+1}|=(n-2)-t(t+1)$ and $|a_t-b_t|=t(t-1)$. Moreover, $f(G)\in[a_t,b_t]$ if and only if $G$ can be obtained from a complete bipartite graph by flipping exactly $t$ edges/nonedges. On the other hand, we show, for all $n$ sufficiently large, that if $m\in[f(n),\binom{n}{3}-f(n)]$, then $m\in F_n$ where $f(n)$ is asymptotically best possible with $f(n)\sim n^{3/2}$ for $n$ even and $f(n)\sim \sqrt{2}n^{3/2}$ for $n$ odd. Furthermore, we determine the graphs with the minimum number of frustrated triangles amongst those with $n$ vertices and $e\leq n^2/4$ edges.
We study inheritance of path-pairability in the Cartesian product of graphs, and prove different (such as additive and multiplicative) inheritance patterns of path-pairability, depending on the size of the Cartesian product. We present path-pairable graph families, that improve the known upper bound on the minimal maximum degree of a path-pairable graph. Further results and open questions about path-pairability are also presented.
We study linkedness of Cartesian product of graphs and prove that the product of an $a$-linked and a $b$-linked graphs is $(a+b-1)$-linked if the graphs are sufficiently large. Further bounds in terms of connectivity are shown. We determine linkedness of product of paths and product of cycles.
A graph on $2k$ vertices is path-pairable if for any pairing of the vertices the pairs can be joined by edge-disjoint paths. The so far known families of path-pairable graphs have diameter of length at most 3. In this paper we present an infinite family of path-pairable graphs with diameter $d(G)=O(\sqrt{n})$ where $n$ denotes the number of vertices of the graph. We prove that our example is extremal up to a constant factor.