Nesetril, Jaroslav

Strong edge colouring conjecture ★★

Author(s): Erdos; Nesetril

A strong edge-colouring of a graph $ G $ is a edge-colouring in which every colour class is an induced matching; that is, any two vertices belonging to distinct edges with the same colour are not adjacent. The strong chromatic index $ s\chi'(G) $ is the minimum number of colours in a strong edge-colouring of $ G $.

Conjecture   $$s\chi'(G) \leq \frac{5\Delta^2}{4}, \text{if $\Delta$ is even,}$$ $$s\chi'(G) \leq \frac{5\Delta^2-2\Delta +1}{4},&\text{if $\Delta$ is odd.}$$


Long rainbow arithmetic progressions ★★

Author(s): Fox; Jungic; Mahdian; Nesetril; Radoicic

For $ k\in \mathbb{N} $ let $ T_k $ denote the minimal number $ t\in \mathbb{N} $ such that there is a rainbow $ AP(k) $ in every equinumerous $ t $-coloring of $ \{ 1,2,\ldots ,tn\} $ for every $ n\in \mathbb{N} $

Conjecture   For all $ k\geq 3 $, $ T_k=\Theta (k^2) $.

Keywords: arithmetic progression; rainbow

Pentagon problem ★★★

Author(s): Nesetril

Question   Let $ G $ be a 3-regular graph that contains no cycle of length shorter than $ g $. Is it true that for large enough~$ g $ there is a homomorphism $ G \to C_5 $?

Keywords: cubic; homomorphism

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