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Edge coloring


Graph Theory » Coloring » Edge coloring

Petersen coloring conjecture ★★★

Author(s): Jaeger

Conjecture   Let $ G $ be a cubic graph with no bridge. Then there is a coloring of the edges of $ G $ using the edges of the Petersen graph so that any three mutually adjacent edges of $ G $ map to three mutually adjancent edges in the Petersen graph.

Keywords: cubic; edge-coloring; Petersen graph

Posted by mdevos
updated November 24th, 2011
2 comments
Graph Theory » Coloring » Edge coloring

Packing T-joins ★★

Author(s): DeVos

Conjecture   There exists a fixed constant $ c $ (probably $ c=1 $ suffices) so that every graft with minimum $ T $-cut size at least $ k $ contains a $ T $-join packing of size at least $ (2/3)k-c $.

Keywords: packing; T-join

Posted by mdevos
updated March 7th, 2007
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Graph Theory » Coloring » Edge coloring

Acyclic edge-colouring ★★

Author(s): Fiamcik

Conjecture   Every simple graph with maximum degree $ \Delta $ has a proper $ (\Delta+2) $-edge-colouring so that every cycle contains edges of at least three distinct colours.

Keywords: edge-coloring

Posted by mdevos
updated March 7th, 2007
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Graph Theory » Coloring » Edge coloring

A generalization of Vizing's Theorem? ★★

Author(s): Rosenfeld

Conjecture   Let $ H $ be a simple $ d $-uniform hypergraph, and assume that every set of $ d-1 $ points is contained in at most $ r $ edges. Then there exists an $ r+d-1 $-edge-coloring so that any two edges which share $ d-1 $ vertices have distinct colors.

Keywords: edge-coloring; hypergraph; Vizing

Posted by mdevos
updated June 23rd, 2009
1 comment
Graph Theory » Coloring » Edge coloring

List colorings of edge-critical graphs ★★

Author(s): Mohar

Conjecture   Suppose that $ G $ is a $ \Delta $-edge-critical graph. Suppose that for each edge $ e $ of $ G $, there is a list $ L(e) $ of $ \Delta $ colors. Then $ G $ is $ L $-edge-colorable unless all lists are equal to each other.

Keywords: edge-coloring; list coloring

Posted by Robert Samal
updated June 12th, 2007
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Graph Theory » Coloring » Edge coloring

Universal Steiner triple systems ★★

Author(s): Grannell; Griggs; Knor; Skoviera

Problem   Which Steiner triple systems are universal?

Keywords: cubic graph; Steiner triple system

Posted by macajova
updated October 5th, 2007
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Graph Theory » Coloring » Edge coloring

Edge list coloring conjecture ★★★

Author(s):

Conjecture   Let $ G $ be a loopless multigraph. Then the edge chromatic number of $ G $ equals the list edge chromatic number of $ G $.

Keywords:

Posted by tchow
updated September 25th, 2008
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Graph Theory » Coloring » Edge coloring

Seymour's r-graph conjecture ★★★

Author(s): Seymour

An $ r $-graph is an $ r $-regular graph $ G $ with the property that $ |\delta(X)| \ge r $ for every $ X \subseteq V(G) $ with odd size.

Conjecture   $ \chi'(G) \le r+1 $ for every $ r $-graph $ G $.

Keywords: edge-coloring; r-graph

Posted by mdevos
updated October 3rd, 2008
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Graph Theory » Coloring » Edge coloring

Goldberg's conjecture ★★★

Author(s): Goldberg

The overfull parameter is defined as follows: \[ w(G) = \max_{H \subseteq G} \left\lceil \frac{ |E(H)| }{ \lfloor \tfrac{1}{2} |V(H)| \rfloor} \right\rceil. \]

Conjecture   Every graph $ G $ satisfies $ \chi'(G) \le \max\{ \Delta(G) + 1, w(G) \} $.

Keywords: edge-coloring; multigraph

Posted by mdevos
updated June 5th, 2013
2 comments
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   $$s\chi'(G) \leq \frac{5\Delta^2}{4}, \text{if $\Delta$ is even,}$$ $$s\chi'(G) \leq \frac{5\Delta^2-2\Delta +1}{4},&\text{if $\Delta$ is odd.}$$

Keywords:

Posted by fhavet
updated March 1st, 2013
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Graph Theory » Coloring » Edge coloring

Black and White Cycle Conjecture ★★★

Author(s): Arthur; Hoffmann-Ostenhof

Conjecture   Let $ G $ be a cubic graph with a nz-$ 4 $-flow, and let each edge $ e \in E(G) $ satisfying $ G-e $ does not have a nz-$ 4 $-flow be colored black, and let each edge $ e \in E(G) $ satisfying $ G-e $ has a nz-$ 4 $-flow be colored white. Then $ G $ contains a white cycle but not a black cycle.


Used terminology: nz-$ 4 $-flow is the abbrevation for nowhere-zero $ 4 $-flow, a cycle is a connected $ 2 $-regular graph, a cycle is called black (white) if each edge of the cycle is colored black (white).

Keywords: 3-edge-coloring; 4-flow; cubic graphs; removable edge

Posted by arthur
updated September 21st, 2026
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Note: Resolved problems from this section may be found in Solved problems.


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