Abstract
In his paper “Fruit salad” (mixed for Paul Erdos) Gyárfás has posed the following conjecture: If each path of a graph spans at most 3-chromatic subgraph then the graph is k-colourable (with a constant k, perhaps with k = 4). We will show that these graphs are colourable with 3 · Illgc ¦V(G)¦⌉ colours for a suitable constant c = 8/7. As a corollary we obtain that every graph G admits a partition of its vertex set V(G) into at most Illgc ¦V(G)¦⌋ subsets Vi for a suitable constant c = 8/7, such that the components of each induced subgraph G[Vi] are spaned by a path.
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References
J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, Macmillan, London and Elsevier, New York (1976).
A. Gyarfas, Fruit Salad, EJC 4 (1997), #R8.
T. Jensen and B. Toft, Graph Coloring Problems, Wiley, 1995.
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Randerath, B., Schiermeyer, I. Chromatic Number of Graphs each Path of which is 3-colourable. Results. Math. 41, 150–155 (2002). https://doi.org/10.1007/BF03322762
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DOI: https://doi.org/10.1007/BF03322762