Generalized chromatic numbers and additive hereditary properties of graphs

Abstract

An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphisms. Let P and Q be additive hereditary properties of graphs. The generalized chromatic number χ Q (P) is defined as follows: χ Q (P) = n iff P ⊆ Q n but P ⊆ Q n−1 . We investigate the generalized chromatic numbers of the well-known properties of graphs I k , O k , W k , S k and D k .

A survey of the Path Partition Conjecture

“…In 1995 Bondy [13] stated a conjecture that is close to the digraph analogue of the PPC, but the PPC for undirected graphs was not mentioned in the literature again until 1997, when it was addressed in Problem 1 of [15] and also in [20]. Since then, several authors have contributed results towards proving the PPC (see [17,18,21,22,25,26,29,30,33,34,45,46,47]) and its digraph analogue, the DPPC (see [1,2,3,4,5,6,8,9,28,31,32,37]). The PPC and DPPC are also discussed in the text books [35] and [10], respectively.…”

“…By a different method, using Theorem 2.8, Broere, Dorfling and Jonck [18] obtained the following bound. A Survey of the PPC 127 In 1995 Bondy [13] formulated a conjecture that is seemingly stronger than the DPPC, requiring λ(D[V i ]) = λ i instead of λ(D[V i ]) ≤ λ i .…”

Fractional Q-coloring of graphs

“…We list several well-known additive hereditary properties Generalized colorings of edges or/and vertices of graphs under restrictions given by graph properties have recently attracted much attention, see e.g. [2,3,4,6,7,8,10] and references therein.…”

Generalized total colorings of graphs

“…Generalized P-vertex colorings and P-chromatic numbers χ P (G) as well as generalized Q-edge colorings and Q-chromatic indices χ ′ Q (G) are analogously defined (see [3,9] for some results). Evidently, these are generalizations of proper vertex colorings and proper edge colorings since χ O (G) = χ(G) and χ ′ O 1 (G) = χ ′ (G).…”