Induced Cycles and Chromatic Number

Induced Cycles and Chromatic Number

We prove that, for any pair of integers k, l ≥ 1, there exists an integerN(k, l) such that every graph with chromatic number at least N(k, l) contains either Kk or an induced odd cycle of length at least 5 or an induced cycle of length at least l. Given a graph with large chromatic number, it is natural to ask whether it must contain induced subgraphs with particular properties. One possibility...

A Note on Chromatic Number and Induced Odd Cycles

An odd hole is an induced odd cycle of length at least 5. Scott and Seymour confirmed a conjecture of Gyárfás and proved that if a graph G has no odd holes then χ(G) 6 22 ω(G)+2 . Chudnovsky, Robertson, Seymour and Thomas showed that if G has neither K4 nor odd holes then χ(G) 6 4. In this note, we show that if a graph G has neither triangles nor quadrilaterals, and has no odd holes of length a...

The b-chromatic number of powers of cycles

Strong Oriented Chromatic Number of Planar Graphs without Short Cycles

Let M be an additive abelian group. An M-strong-oriented coloring of an oriented graph G is a mapping φ from V (G) to M such that φ(u) 6= φ(v) whenever −→uv is an arc in G and φ(v)−φ(u) 6= −(φ(t)−φ(z)) whenever −→uv and zt are two arcs in G. The strong oriented chromatic number of an oriented graph is the minimal order of a group M such that G has an M-strong-oriented coloring. This notion was ...

Cycles in triangle-free graphs of large chromatic number

More than twenty years ago Erdős conjectured [4] that a triangle-free graph G of chromatic number k ≥ k0(ε) contains cycles of at least k2−ε different lengths as k →∞. In this paper, we prove the stronger fact that every triangle-free graph G of chromatic number k ≥ k0(ε) contains cycles of 1 64 (1− ε)k 2 log k4 consecutive lengths, and a cycle of length at least 14 (1− ε)k 2 log k. As there ex...