Triangle-free graphs with large chromatic numbers

Proper conflict-free list-coloring, odd minors, subdivisions, and layered treewidth

Induced subgraphs of graphs with large chromatic number. IV. Consecutive holes

The present paper is devoted to the study of the properties of distance graphs in Euclidean space. We prove, in particular, the existence of distance graphs with chromatic number of exponentially large dimension and without cliques of dimension 6. In addition, under a given constraint on the cardinality of the maximal clique, we search for distance graphs with extremal large values of the chromatic number. The resulting estimates are best possible within the framework of the proposed method, which combines probabilistic techniques with the linear-algebraic approach.

Um resultado de Erdos demonstra a existência de grafos com número cromático e cintura arbitrariamente grandes. Temos então que um clique suficientemente grande contém um grafo com número cromático e cintura grandes como subgrafo, porém muitos grafos de interesse não necessariamente contém cliques grandes, então é interessante encontrar outra condição que garanta a existência de subgrafos com número cromático e cintura grandes. Uma conjectura de Erdos e Hajnal diz que todo grafo com número cromático suficientemente grande contém um subgrafo com número cromático e cintura grandes. O objetivo deste trabalho é estudar tal conjectura. O texto começa com uma breve apresentação de construções livres de triângulos. Em particular, é demonstrada uma construção de Codenotti, Pudlák e Resta, por meio de planos projetivos. O tópico principal do texto começa com uma demonstração de Rödl de que todo grafo com número cromático suficientemente grande contém um subgrafo livre de triângulos e com número cromático grande. Em sequência, apresentaremos uma demonstração de que grafos com número cromático suficientemente grande contém algum circuito ímpar grande. Apresentaremos também um resultado de Mohar e Wu, que demonstra que a família dos grafos de Kneser respeita a conjectura de Erdos e Hajnal. Outro resultado apresentado é de Gábor Tardos, demonstrando que a família dos shift graphs respeita a conjectura de Erdos e Hajnal. E por fim apresentaremos alguns breves resultados sobre os type graphs, mostrando casos que respeitam a conjectura de Erdos e Hajnal.

We study the computational efficiency of approaches, based on Hilbert's Nullstellensatz, which use systems of linear equations for detecting non-colorability of graphs having large girth and chromatic number. We show that for every non-k-colorable graph with n vertices and girth g > 4k, the algorithm is required to solve systems of size at least n Ω(g) in order to detect its non-k-colorability.

A cycle with four blocks $C(k_{1}, k_{2},k_{3},k_{4})$ is an oriented cycle formed of four blocks of lengths $k_{1}, k_{2}, k_{3}$ and $k_{4}$ respectively. Recently, Cohen et al. conjectured that for every positive integers $k_{1}, k_{2}, k_{3}, k_{4}$, there is an integer $g(k_{1},k_{2},k_{3},k_{4})$ such that every strongly connected digraph $D$ containing no subdivisions of $C(k_{1},k_{2},k_{3},k_{4})$ has a chromatic number at most $g(k_{1},k_{2},k_{3},k_{4})$. This conjecture is confirmed by Cohen et al. for the case of $C(1,1,1,1)$ and by Al-Mniny for the case of $C(k_1,1,1,1)$. In this paper, we affirm Cohen et al.'s conjecture for the case where $k_2=k_4=1$, namely $g(k_1,1,k_3,1) =O({(k_1+k_3)}^2)$. Moreover, we show that if in addition $D$ is Hamiltonian, then the chromatic number of $D$ is at most $6k$, with $k=\textrm{max}\{k_1,k_3\}.$

Let Q (n, χ) denote the minimum clique size an n-vertex graph can have if its chromatic number is χ. Using Ramsey graphs we give an exact, albeit implicit, formula for the case χ ≥ (n + 3)/2.

Triangle-free graphs whose independence number equals the degree

Induced subgraphs of graphs with large chromatic number. XIII. New brooms

Circular chromatic numbers of Mycielski's graphs

Distributed coloring algorithms for triangle-free graphs

We construct families of circles in the plane such that their tangency graphs have arbitrarily large girth and chromatic number. This provides a strong negative answer to Ringel’s circle problem (1959). The proof relies on a (multidimensional) version of Gallai’s theorem with polynomial constraints, which we derive from the Hales–Jewett theorem and which may be of independent interest.

Sparse Ramsey theorems for graphs originated with investigations of graphs having large chromatic number and high girth (where the girth of a graph is the length of the smallest cycle in G). Note that this can be viewed as a special kind of restricted graph Ramsey problem.

P. Erdős and A. Hajnal asked the following question. Does there exist a constant e>0 with the following property: If every subgraphH of a graphG can be made bipartite by the omission of at most e|H| edges where |H| denotes the number of vertices ofH thenx(H) ≦ 3.

Triangle-free graphs and forbidden subgraphs