A topological graph is a graph drawn in the plane so that its vertices are represented by points, and its edges are represented by Jordan curves connecting the corresponding points, with the property that any two curves have at most one point in common. We define two canonical classes of topological complete graphs, and prove that every topological complete graph with n vertices has a canonical...

Classic results from extremal graph theory state that if certain graphs are made large enough, unavoidable substructures appear. Here we will cover this type of problem for specific graphs when these substructures are certain trees or forests. After giving a summary on related results, the following two extremal main problems are presented: For a given family of same-order trees including the s...

Ramsey proved that for every positive integer $n$, sufficiently large graph contains an induced $K_n$ or $\overline{K}_n$. Among the many extensions of Ramsey's Theorem there is analogue connected graphs: $K_n$, $K_{1,n}$, $P_n$. In this paper, we establish 2-connected graphs. particular, prove exceeding two, one following as subgraph: a subdivision $K_{2,n}$, $K_{2,n}$ with edge between two ve...

We say that a word w on a totally ordered alphabet avoids the word v if there are no subsequences in w order-equivalent to v. In this paper we suggest a new approach to the enumeration of words on at most k letters avoiding a given pattern. By studying an automaton which for fixed k generates the words avoiding a given pattern we derive several previously known results for these kind of problem...

In this paper we study pattern avoidance for affine permutations. In particular, we show that for a given pattern p, there are only finitely many affine permutations in S̃n that avoid p if and only if p avoids the pattern 321. We then count the number of affine permutations that avoid a given pattern p for each p in S3, as well as give some conjectures for the patterns in S4. This paper is just ...

We say that a word w on a totally ordered alphabet avoids the word v if there are no subsequences in w order-equivalent to v. In this paper we suggest a new approach to the enumeration of words on at most k letters avoiding a given pattern. By studying an automaton which for fixed k generates the words avoiding a given pattern we derive several previously known results for these kind of problem...

Motivated by a geometrical Thue-type problem, we introduce a new variant of the classical pattern avoidance in words, where jumping over a letter in the pattern occurrence is allowed. We say that pattern p ∈ E+ occurs with jumps in a word w = a1a2 . . . ak ∈ A+, if there exist a non-erasing morphism f from E∗ to A∗ and a sequence (i1, i2, . . . , il) satisfying ij+1 ∈ {ij + 1, ij + 2} for j = 1...