Forbidden Subgraphs in Connected Graphs

Given a set $\xi=\{H_1,H_2,...\}$ of connected non acyclic graphs, a $\xi$-free graph is one which does not contain any member of $$ as copy. Define the excess of a graph as the difference between its number of edges and its number of vertices. Let {\gr{W}}_{k,\xi} be theexponential generating function (EGF for brief) of connected $\xi$-free graphs of excess equal to $k$ ($k \geq 1$). For each fixed $\xi$, a fundamental differential recurrence satisfied by the EGFs {\gr{W}}_{k,\xi} is derived. We give methods on how to solve this nonlinear recurrence for the first few values of $k$ by means of graph surgery. We also show that for any finite collection $\xi$ of non-acyclic graphs, the EGFs {\gr{W}}_{k,\xi} are always rational functions of the generating function, $T$, of Cayley's rooted (non-planar) labelled trees. From this, we prove that almost all connected graphs with $n$ nodes and $n+k$ edges are $\xi$-free, whenever $k=o(n^{1/3})$ and $|\xi| < \infty$ by means of Wright's inequalities and saddle point method. Limiting distributions are derived for sparse connected $\xi$-free components that are present when a random graph on $n$ nodes has approximately $\frac{n}{2}$ edges. In particular, the probability distribution that it consists of trees, unicyclic components, $...$, $(q+1)$-cyclic components all $\xi$-free is derived. Similar results are also obtained for multigraphs, which are graphs where self-loops and multiple-edges are allowed

Master index

Pla general, del mural ceràmic que decora una de les parets del vestíbul de la Facultat de Química de la UB. El mural representa diversos símbols relacionats amb la química

Core Structures in Random Graphs and Hypergraphs

The k-core of a graph is its maximal subgraph with minimum degree at least k. The study of k-cores in random graphs was initiated by Bollobás in 1984 in connection to k-connected subgraphs of random graphs. Subsequently, k-cores and their properties have been extensively investigated in random graphs and hypergraphs, with the determination of the threshold for the emergence of a giant k-core, due to Pittel, Spencer and Wormald, as one of the most prominent results. In this thesis, we obtain an asymptotic formula for the number of 2-connected graphs, as well as 2-edge-connected graphs, with given number of vertices and edges in the sparse range by exploiting properties of random 2-cores. Our results essentially cover the whole range for which asymptotic formulae were not described before. This is joint work with G. Kemkes and N. Wormald. By defining and analysing a core-type structure for uniform hypergraphs, we obtain an asymptotic formula for the number of connected 3-uniform hypergraphs with given number of vertices and edges in a sparse range. This is joint work with N. Wormald. We also examine robustness aspects of k-cores of random graphs. More specifically, we investigate the effect that the deletion of a random edge has in the k-core as follows: we delete a random edge from the k-core, obtain the k-core of the resulting graph, and compare its order with the original k-core. For this investigation we obtain results for the giant k-core for Erdős-Rényi random graphs as well as for random graphs with minimum degree at least k and given number of vertices and edges