Fractional Thue chromatic number of graphs

Abstract

a b s t r a c tThis paper introduces the concept of the fractional Thue chromatic number of graphs and studies the relation between the fractional Thue chromatic number and the Thue chromatic number. We determine the fractional Thue chromatic number of all paths, all trees with no vertices of degree two, and all cycles, except C 10 , C 14 , C 17 . As a consequence, we prove that if G is a path or a tree with no degree two vertices, then its fractional Thue chromatic number equals its Thue chromatic number. On the other hand, we show that there are trees and cycles whose fractional Thue chromatic numbers are strictly less than their Thue chromatic numbers.

The Thue choice number versus the Thue chromatic number of graphs

“…the Thue chromatic index, π ′ (G) -see [11], [30]; the Thue choice index, π ′ l (G) -see [26]; the facial Thue chromatic index, π ′ f (G) -see [26], [38], [39], [62]; the facial Thue choice index, π ′ f l (G) -see [26], [55], [59]), while the Thue graph parameter connected with nonrepetitive vertex colourings or total colourings will be called number and abbreviated without apostrophe (e.g. the Thue chromatic number, π(G) -see [11], [12], [15], [26], [27], [28], [29], [30], [34], [35], [40], [41], [53], [54], [58]; the Thue choice number, π l (G) -see [24], [26], [32] 3 , [42]; the facial Thue choice number, π f l (G) -see [26], [56]; the facial Thue chromatic number, π f (G) -see [5], [26], [33], [34], unhapilly, with the same abbreviation like the fractional Thue chromatic number, π f (G) -see [40], [64]; for the Thue parameters related to total Thue colourings see [43], [60]). We will also follow this idea and use the abbreviation π(G) for the Thue chromatic number and π l (G) for the Thue choice number of a graph G. Except of the few notation defined throughout the paper we will use the standard terminology according to Bondy and Murty [10].…”