Fractional ({ P},{ Q})-total list colorings of graphs

Abstract

Let r, s ∈ N, r ≥ s, and P and Q be two additive and hereditary graph properties. A (P, Q)-total (r, s)-coloring of a graph G = (V, E) is a coloring of the vertices and edges of G by s-element subsets of Z r such that for each color i, 0 ≤ i ≤ r − 1, the vertices colored by subsets containing i induce a subgraph of G with property P, the edges colored by subsets containing i induce a subgraph of G with property Q, and color sets of incident vertices and edges are disjoint. The fractional (P, Q)-total chromatic number χ ′′ f,P,Q (G) of G is defined as the infimum of all ratios r/s such that G has a (P, Q)-total (r, s)-coloring.

(P,Q)-Total (r,s)

“…for every fractional (P , Q)-total coloring of G. This inequality implies the following theorem which is a generalization of Theorem 6 in [7] and also of Proposition 2 in [5].…”

“…Theorems 8 and 9 in [7] provide graph classes such that the lower bound is attained. In Sections 4.1 and 4.2 we will show that some regular graphs and complete multipartite graphs have this property too.…”

“…We know from Theorems 8 and 9 in [7] that this bound is tight for S 2 ∩ D 1 ⊆ P , Q ⊆ D 1 if the graph is a cycle, i.e., if r = 2, as well as for K n ̸ ∈ P ⊇ O 1 , P n ∈ Q ⊆ D 1 or K 1,n−1 ∈ Q ⊆ D 1 if the graph is a complete graph K n with n ≥ 3, i.e., if r = n − 1. In the following we give constructions proving the tightness of this bound also for r = n − 2 and r = n − 3.…”

“…Bounds and exact values for χ ′′ f ,P ,Q (G) for some properties P and Q and some classes of graphs are obtained in [7]. In [6] it was shown that the following definition of the fractional (P , Q)-total chromatic number is equivalent to the one given above.…”