Clique immersion in graphs without fixed bipartite graph

A graph $G$ contains $H$ as an \emph{immersion} if there is an injective mapping $\phi: V(H)\rightarrow V(G)$ such that for each edge $uv\in E(H)$, there is a path $P_{uv}$ in $G$ joining vertices $\phi(u)$ and $\phi(v)$, and all the paths $P_{uv}$, $uv\in E(H)$, are pairwise edge-disjoint. An analogue of Hadwiger's conjecture for the clique immersions by Lescure and Meyniel states that every graph $G$ contains $K_{\chi(G)}$ as an immersion. We consider the average degree condition and prove that for any bipartite graph $H$, every $H$-free graph $G$ with average degree $d$ contains a clique immersion of order $(1-o(1))d$, implying that Lescure and Meyniel's conjecture holds asymptotically for graphs without fixed bipartite graph.Comment: 2 figure

Clique minors in graphs with a forbidden subgraph

The classical Hadwiger conjecture dating back to 1940's states that any graph of chromatic number at least $r$ has the clique of order $r$ as a minor. Hadwiger's conjecture is an example of a well studied class of problems asking how large a clique minor one can guarantee in a graph with certain restrictions. One problem of this type asks what is the largest size of a clique minor in a graph on $n$ vertices of independence number $\alpha(G)$ at most $r$. If true Hadwiger's conjecture would imply the existence of a clique minor of order $n/\alpha(G)$. Results of Kuhn and Osthus and Krivelevich and Sudakov imply that if one assumes in addition that $G$ is $H$-free for some bipartite graph $H$ then one can find a polynomially larger clique minor. This has recently been extended to triangle free graphs by Dvo\v{r}\'ak and Yepremyan, answering a question of Norin. We complete the picture and show that the same is true for arbitrary graph $H$, answering a question of Dvo\v{r}\'ak and Yepremyan. In particular, we show that any $K_s$-free graph has a clique minor of order $c_s(n/\alpha(G))^{1+\frac{1}{10(s-2) }}$, for some constant $c_s$ depending only on $s$. The exponent in this result is tight up to a constant factor in front of the $\frac{1}{s-2}$ term.Comment: 11 pages, 1 figur

Minors in expanding graphs

Extending several previous results we obtained nearly tight estimates on the maximum size of a clique-minor in various classes of expanding graphs. These results can be used to show that graphs without short cycles and other H-free graphs contain large clique-minors, resolving some open questions in this area

Limits of degeneracy for colouring graphs with forbidden minors

Motivated by Hadwiger's conjecture, Seymour asked which graphs $H$ have the property that every non-null graph $G$ with no $H$ minor has a vertex of degree at most $|V(H)|-2$. We show that for every monotone graph family $\mathcal{F}$ with strongly sublinear separators, all sufficiently large bipartite graphs $H \in \mathcal{F}$ with bounded maximum degree have this property. None of the conditions that $H$ belongs to $\mathcal{F}$, that $H$ is bipartite and that $H$ has bounded maximum degree can be omitted.Comment: 22 page

Combinatorics, Probability and Computing

One of the exciting phenomena in mathematics in recent years has been the widespread and surprisingly effective use of probabilistic methods in diverse areas. The probabilistic point of view has turned out to b

Expanding Graphs and Balanced Separators

Ένα γράφημα ονομάζεται εξαπλωτής αν είναι αραιό αλλά ταυτόχρονα έχει ισχυρές ιδιότητες συνεκτικότητας. Οι εξαπλωτές είναι μία κατηγορία γραφημάτων η οποία, κυρίως λόγω των πολλών εφαρμογών τους σε διαφορετικά πεδία των μαθηματικών, έχουν μελετηθεί εκτενώς. Ο στόχος αυτής της εργασίας είναι να αναλύσουμε τη σύνδεση των εξαπλωτών με άλλες έννοιες της θεωρίας γραφημάτων, και να μελετήσουμε τις δομές που μπορούμε να βρούμε σε αυτούς. Συγκεκριμένα, θα επικεντρωθούμε στους ισορροπημένους διαχωριστές και πώς αυτοί συνδέονται με τους εξαπλωτές. Επιπλέον θα δούμε πιο σύντομα, πώς οι ιδιοτιμές του πίνακα γειτνίασης ενός γραφήματος συνδέονται με την εξάπλωσή του αλλά και με άλλες ιδιότητές του. Τέλος, θα ασχοληθούμε ιδιαίτερα με τα ελάσσονα γραφήματα ενός εξαπλωτή.A graph is an expander if it is sparse and has strong connectivity properties. Expanders are widely studied graphs, mainly due to their numerous applications in many different mathematical fields. The purpose of this thesis is to analyze the connections between expanders and other notions of graph theory, and study their substructures. Specifically, we will focus on the connection of balanced separators and expanders and provide an introduction on how the expansion of a graph is connected to the eigenvalues of its adjacency matrix. We will also study in detail the minors one can find in expanders

Extremal Graph Theory: Basic Results

Η παρούσα διπλωματική εργασία έχει σκοπό να παρουσιάσει μία σφαιρική εικόνα της θεωρίας των ακραίων γραφημάτων, διερευνώντας κοινές τεχνικές και τον τρόπο που εφαρμόζονται σε κάποια από τα πιο διάσημα αποτελέσματα του τομέα. Το πρώτο κεφάλαιο είναι μία εισαγωγή στο θέμα και κάποιοι προαπαιτούμενοι ορισμοί και αποτελέσματα. Το δεύτερο κεφάλαιο αφορά υποδομές πυκνών γραφημάτων και εστιάζει σε σημαντικά αποτελέσματα όπως είναι το θεώρημα του Turán, το λήμμα κανονικότητας του Szemerédi και το θεώρημα των Erdős-Stone-Simonovits. Το τρίτο κεφάλαιο αφορά υποδομές αραιών γραφημάτων και ερευνά συνθήκες που εξαναγκάζουν ένα γράφημα που περιέχει ένα δοθέν έλασσον ή τοπολογικό έλασσον. Το τέταρτο και τελευταίο κεφάλαιο είναι μία εισαγωγή στην θεωρία ακραίων r-ομοιόμορφων υπεργραφημάτων και περιέχει αποτελέσματα που αφορούν συνθήκες οι οποίες τα εξαναγκάζουν να περιέχουν πλήρη r-γραφήματα και Χαμιλτονιανούς κύκλους.In this thesis, we take a general overview of extremal graph theory, investigating common techniques and how they apply to some of the more celebrated results in the field. The first chapter is an introduction to the subject and some preliminary definitions and results. The second chapter concerns substructures in dense graphs and focuses on important results such as Turán’s theorem, Szemerédi’s regularity lemma and the Erdős-Stone-Simonovits theorem. The third chapter concerns substructures in sparse graphs and investigates conditions which force a graph to contain a certain minor or topological minor. The fourth and final chapter is an introduction to the extremal theory of r-uniform hypergraphs and consists of a presentation of results concerning the conditions which force them to contain a complete r-graph and a Hamiltonian cycle