×

zbMATH — the first resource for mathematics

Fractionally colouring total graphs. (English) Zbl 0795.05056
To any simple graph \(G=(V,E)\) with maximum vertex degree \(\Delta\) we form a total graph with stable set \(T\) on the vertex set \(V(G) \cup E(G)=F\). It is shown that a feasible solution of the minimum in the linear program \[ \min \left\{ w\bar 1:w \geq 0,\sum_{u \in T} w_ T \geq 1,\;u \in F,\;T \in {\mathcal S} \right\} \] is bounded above by \(\Delta+2\), where \({\mathcal S}\) is the family of total stable sets of \(G\).
Reviewer: J.Fiamcik

MSC:
05C15 Coloring of graphs and hypergraphs
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
90C10 Integer programming
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] M. Behzad: Graphs and their chromatic numbers,Doctoral Thesis (Michigan State University)., (1965).
[2] J. Edmonds: Maximum matching and a polyhedron with 0,1 vertices,Journal of Research of the National Bureau of Standards (B) 69 (1965), 125-130. · Zbl 0141.21802
[3] D. R. Fulkerson: Anti-blocking polyhedra,J. Combinatorial Theory B 12 (1972). 50-71. · Zbl 0227.05015
[4] H. R. Hind: An upper bound on the total chromatic number.,Graphs and Combinatorics 6 (1990), 153-158. · Zbl 0725.05043
[5] J. Ryan: Fractional total colouring,Discrete Appl. Math. 27 (1990), 287-292. · Zbl 0731.05018
[6] N. Vijayaditya: On the total chromatic number of a graph,J. London Math. Soc. (2)3 (1971), 405-408. · Zbl 0223.05103
[7] V. G. Vizing: Some unsolved problems in graph theory (in Russian),Uspekhi Math. Nauk. 23 (1968), 117-134.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.