Graph Theory » Coloring » Vertex coloring

Melnikov's valency-variety problem ★

Author(s): Melnikov

Problem   The valency-variety $ w(G) $ of a graph $ G $ is the number of different degrees in $ G $. Is the chromatic number of any graph $ G $ with at least two vertices greater than $$\ceil{ \frac{\floor{w(G)/2}}{|V(G)| - w(G)} } ~ ?$$

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Posted by asp
updated March 3rd, 2013
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Graph Theory

Do any three longest paths in a connected graph have a vertex in common? ★★

Author(s): Gallai

Conjecture   Do any three longest paths in a connected graph have a vertex in common?

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Posted by fhavet
updated October 14th, 2023
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Graph Theory » Coloring

Coloring the union of degenerate graphs ★★

Author(s): Tarsi

Conjecture   The union of a $ 1 $-degenerate graph (a forest) and a $ 2 $-degenerate graph is $ 5 $-colourable.

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Posted by fhavet
updated March 3rd, 2013
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Graph Theory

Arc-disjoint strongly connected spanning subdigraphs ★★

Author(s): Bang-Jensen; Yeo

Conjecture   There exists an ineteger $ k $ so that every $ k $-arc-connected digraph contains a pair of arc-disjoint strongly connected spanning subdigraphs?

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Posted by fhavet
updated March 2nd, 2013
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Graph Theory » Directed Graphs

Arc-disjoint out-branching and in-branching ★★

Author(s): Thomassen

Conjecture   There exists an integer $ k $ such that every $ k $-arc-strong digraph $ D $ with specified vertices $ u $ and $ v $ contains an out-branching rooted at $ u $ and an in-branching rooted at $ v $ which are arc-disjoint.

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Posted by fhavet
updated March 2nd, 2013
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Graph Theory » Coloring » Edge coloring

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

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Posted by fhavet
updated March 1st, 2013
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