Counterexample to Babai's lonely colour conjecture

Abstract

Motivated by colouring minimal Cayley graphs, in 1978, Babai conjectured that no-lonely-colour graphs have bounded chromatic number. We disprove this in a strong sense by constructing graphs of arbitrarily large girth and chromatic number that have a proper edge-colouring in which each cycle contains no colour exactly once.

Figures (3)

• Figure 1: On the left, we see a 3-uniform hypergraph $H$ with three edges and a labelling for each edge. The walk $W=dAcBfCd$ is the unique closed walk in $H$. On the right, we see the multigraph $G$ corresponding to $W,$ which is bridgeless. Thus, the hypergraph $H$ is tranquil.
• Figure 2: Additional to the independent set $T$, for every edge $F_i \in E( H )$ we add a copy $G_i$ of $G_{g,k-1}$ and a matching between it and the corresponding vertices in $T.$
• Figure 3: The cycle $C$ in $G_{g,k}$ drawn in green corresponds to the walk $W_{H}( C ) = a F_1 b F_2 c F_3 d F_4 e F_5 f F_3 a$ in the hypergraph $H$.

Theorems & Definitions (16)

• Theorem 0
• Conjecture 0: Babai babai1978chromatic
• Theorem 1
• Lemma 2
• proof
• Theorem 3: Prömel, Voigt PV88PV90
• Theorem 4
• Lemma 5
• proof
• Lemma 6