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It is shown that under some fairly general conditions the Overfull Conjecture about the chromatic index of a graph G implies the Conformability Conjectures about the total chromatic number of G. Expand

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#### References

SHOWING 1-10 OF 27 REFERENCES

SOME UNSOLVED PROBLEMS IN GRAPH THEORY

- Mathematics
- 1968

CONTENTSIntroduction § 1. Fundamental concepts § 2. Isomorphism problems § 3. Metric questions § 4. Thickness and genus of graph § 5. Colouring problems § 6. Parts with given propertiesReferences

A Bound on the Strong Chromatic Index of a Graph,

- Computer Science, Mathematics
- J. Comb. Theory, Ser. B
- 1997

We show that the strong chromatic index of a graph with maximum degree�; is at most (2��)�2, for some�>0. This answers a question of Erdo�s and Ne�et�il.

Some upper bounds on the total and list chromatic numbers of multigraphs

- Mathematics, Computer Science
- J. Graph Theory
- 1992

In this paper we discuss some estimates for upper bounds on a number of chromatic parameters of a multigraph. In particular, we show that the total chromatic number for an n-order multigraph exceeds… Expand

Total Colourings of Graphs

- Mathematics
- 1996

Basic terminology and introduction.- Some basic results.- Complete r-partite graphs.- Graphs of low degree.- Graphs of high degree.- Classification of type 1 and type 2 graphs.- Total chromatic… Expand

Fractionally colouring total graphs

- Mathematics, Computer Science
- Comb.
- 1993

It is shown that for any simple graphG, V (G)ϒE(G) can be fractionally coloured with Δ+2 colours. Expand

Colouring a graph frugally

- Mathematics, Computer Science
- Comb.
- 1997

We prove that any graph with maximum degree Δ sufficiently large, has a proper vertex colouring using Δ+1 colours such that each colour class appears at most log8 Δ times in the neighbourhood of any… Expand

The Probabilistic Method

- Computer Science
- SODA
- 1992

A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored. Expand

Further algorithmic aspects of the local lemma

- Computer Science
- STOC '98
- 1998

This is the author's version of the work and it is posted here by permission of ACM for your personal use. Expand

WEIGHTED SUMS OF CERTAIN DEPENDENT RANDOM VARIABLES

- Mathematics
- 1967

1. Let be a probability space and,be an increasing family of sub o'-fields of(we put(c) Let (xn)n=1, 2, •c be a sequence of bounded martingale differences on , that is,xn(ƒÖ) is bounded almost surely… Expand

An Introduction To Probability Theory And Its Applications

- Computer Science
- 1950

A First Course in Probability (8th ed.) by S. Ross is a lively text that covers the basic ideas of probability theory including those needed in statistics. Expand