A solution to a colouring problem of P. Erdős

Chords in longest cycles

“…For each pleasant vertex we call one of its pleasant neighbors very pleasant. By the Fleischner-Stiebitz theorem [7] we can select the very pleasant neighbors in such a way that no two of them are consecutive on C. To see that we form a so-called cycle-plus-triangles graph from the cycle C by adding a triangle consisting of the three pleasant neighbors of each pleasant vertex. The Fleischner-Stiebitz theorem implies that this graph is 3-colorable, and we now let the very pleasant neighbors be the pleasant neighbors of color 1, say.…”

“…If v has degree at least 4, then all neighbors of v have degree 3. For every component Q in G − E(C) we select three vertices x Q , y Q , z Q in V (Q) ∩ V (C) such that as many as possible have degree 3 in G. It is easy to see that all of x Q , y Q , z Q have degree 3 unless Q has 6 vertices x Q , y Q , z Q , u, v, w such that x Q , y Q , z Q , w are in C, u, v are outside C, u is joined to x Q , y Q , w, and v is joined to z Q , y Q , w. We now apply the Fleischner-Stiebitz theorem [7] to the cycle-plus-triangles graph obtained from C by adding the three edges…”

“…This idea was carried further in [16] where the chord-conjecture was verified for the class of cubic 3-connected graphs. In that proof a red-independent, green-dominating set (in an appropriate auxiliary graph) was found using the Fleischner-Stiebitz theorem [7] saying that every cycle-plus-triangles graph has chromatic number 3.…”

The graph polynomial and the number of proper vertex coloring

“…Fleischner and Stiebitz [3] proved that in every ( Cycle and triangles' graph there is an odd number of Eulerian orientations. They used this result and the 'Graph polynomial method' presented in [I], to prove that such graphs are 3-colorable.…”

General Parity Result and Cycle‐Plus‐Triangles Graphs

“…In [1] the authors used the method of [2], which was later explained as application of Combinatorial Nullstellensatz in the main survey by Alon [5].…”