Steinberg's conjecture is false

Steinberg conjectured in 1976 that every planar graph with no cycles of length four or five is 3-colorable. We disprove this conjecture

A note on 3-choosability of planar graphs without certain cycles

AbstractSteinberg asked whether every planar graph without 4 and 5 cycles is 3-colorable. Borodin, and independently Sanders and Zhao, showed that every planar graph without any cycle of length between 4 and 9 is 3-colorable. We improve this result by showing that every planar graph without any cycle of length 4, 5, 6, or 9 is 3-choosable

Chords of longest circuits of graphs

This thesis is on a long standing open conjecture proposed by one of the most prominent mathematicians, Dr. C. Thomassen: Every longest circuit of 3-connected graph has a chord. In 1987, C. Q. Zhang proved that every longest circuit of a 3-connected planar graph G has a chord if G is cubic or if the minimum degree is at least 4. In 1997, Carsten Thomassen proved that every longest circuit of 3-connected cubic graph has a chord.;In this dissertation, we prove the following three independent partial results: (1) Every longest circuit of a 3-connected graph embedded in a projective plane with minimum degree at least has a chord (Theorem 2.3.1). (2) Every longest circuit of a 3-connected cubic graph has at least two chords. Furthermore if the graph is also a planar, then every longest circuit has at least three chords (Theorem 3.2.6, 3.2.7). (3) Every longest circuit of a 4-connected graph embedded in a torus or Klein bottle has a chord.;We get these three independent results with three totally different approaches: Connectivity (Tutte circuit), second Hamilton circuit, and charge and discharge methods