Goldberg's conjecture

Importance: High ✭✭✭
Author(s): Goldberg, Mark K.
Subject: Graph Theory
» Coloring
» » Edge coloring
Recomm. for undergrads: no
Posted by: mdevos
on: October 4th, 2008

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) \} $.

This important problem remains open despite considerable attention. The same conjecture was independently discovered by Andersen and Seymour.

Vizing's Theorem, one of the cornerstones of graph colouring, shows that $ \chi'(G) \le \Delta(G) + 1 $ for every simple graph $ G $. So, in particular, every simple graph satisfies Goldberg's conjecture. Graphs with parallel edges need not satisfy Vizing's bound. For instance, if $ G $ is the graph obtained from a triangle by adding an extra $ k-1 $ edges in parallel with each existing one, then $ \Delta(G) = 2k $ but $ \chi'(G) = 3k $. More generally, if $ H $ is a subgraph of $ G $, then every colour can appear on at most $ \lfloor \frac{1}{2}|V(H)| \rfloor $ edges of $ H $, so $ \chi'(G) \ge |E(H)| / \lfloor \tfrac{1}{2} |V(H)| \rfloor  $. Thus, $ w(G) $, our overfull parameter, is a natural lower bound on $ \chi'(G) $, and Goldberg's conjecture asserts that whenever $ \chi'(G) $ exceeds $ \Delta(G)+1 $, then it is equal to this lower bound.

Although the statement of the conjecture may appear to be the most natural formulation, there are a couple of related conjectures with similar lower bounds. For instance, Seymour's r-graph conjecture is equivalent to the statement that $ \chi'(G) \le \max \{\Delta(G), w(G) \} + 1 $. Goldberg also conjectured that $ \chi'(G) \le \max\{ \Delta(G), w(G) + 1\} $.

In addition to simple graphs, Goldberg's Conjecture is known to hold for any graph $ G $ which satisfies one of the following

    \item $ \Delta(G) \le 11 $ \item $ G $ has no minor isomorphic to $ K_5 $ minus an edge. \item $ \Delta(G) $ is sufficiently large in comparison with $ |V(G)| $.

$ \quad $

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*[G] M. K. Goldberg, Multigraphs with a chromatic index that is nearly maximal. (Russian) A collection of articles dedicated to the memory of Vitaliĭ Konstantinovič Korobkov. Diskret. Analiz No. 23 (1973), 3--7, 72. MathSciNet

* indicates original appearance(s) of problem.

Latest Developments

Stiebitz et al(2006), Yu(2008) and Kurt(2009) has separately shown $ \chi'(G) \geq \Delta+\sqrt{\Delta/2} $ implies Goldberg Conjecture. While Yu's method gives a methodological approach to the general problem, Kurt provides a very short and elementary proof.

Scheide (2008) has shown $ \chi'(G)>\frac{15}{14}\Delta+\frac{12}{14} $ implies the Goldberg Conjecture.

Kurt(2009) has shown $ \chi'(G)>\frac{17}{16}\Delta+\frac{14}{16} $ implies the Goldberg Conjecture.

Different Goldberg Conjecture?


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