Generalized sum list colorings of graphs

Abstract

A (graph) property P is a class of simple finite graphs closed under isomorphisms. In this paper we consider generalizations of sum list colorings of graphs with respect to properties P. If to each vertex v of a graph G a list L(v) of colors is assigned, then in an (L, P)-coloring of G every vertex obtains a color from its list and the subgraphs of G induced by vertices of the same color are always in P. The P-sum choice number χ P sc (G) of G is the minimum of the sum of all list sizes such that, for any assignment L of lists of colors with the given sizes, there is always an (L, P)-coloring of G. We state some basic results on monotonicity, give upper bounds on the Psum choice number of arbitrary graphs for several properties, and determine the P-sum choice number of specific classes of graphs, namely, of all complete graphs, stars, paths, cycles, and all graphs of order at most 4.

Hypergraph operations preserving sc-greediness

“…Several generalizations, e.g. sumlist colourings in which colour classes need not be edgeless were investigated in [8,9,16]. Effectively computable upper bound on the minimum sum of the list sizes for graphs we find in [14].…”

Sum-list colouring of unions of a hypercycle and a path with at most two vertices in common

“…A colouring of a graph G with respect to a class R of graphs is an assignment of colours to the vertices of G so that each colour class induces in G a graph in R. Note that, immediately by definitions, an assignment of colours to vertices of G is a colouring of G with respect to an induced hereditary class R of graphs if and only if it is a proper colouring of H(G, R). The problem of sum-list colouring of graphs with respect to induced hereditary class R of graphs was introduced in 2016 (see [6]) and it was also investigated in [5,11,15]. Given a function f : V (G) −→ N and an induced hereditary class R of graphs, a graph G is (f, R)-choosable if for every f -assignment L there is an L-colouring of G with respect to R. The R-sum-choice-number χ R sc (G) of G is defined to be the minimum of v∈V (G) f (v) taken over all f such that G is (f, R)-choosable.…”

“…It is known that paths, cycles, trees, complete graphs, and all graphs on at most four vertices are sc-greedy. All sc-greedy graphs on five vertices were determined by Lastrina [16] and on six vertices by Kemnitz et al [13]. Moreover, all sc-greedy complete multipartite graphs [12], wheels [14], and broken wheels [8] were determined.…”

“…Isaak [10] proved that block graphs are sc-greedy. Kemnitz et al [13] considered sc-greediness of the Cartesian product, the direct product, the strong product, and the lexicographic product of two graphs.…”

“…In [5,6,11,15] some generalization of the sum-list colourability concept has been investigated. In this variant the graphs induced by the vertices of the same colour belong to some specific fixed class of graphs (not necessarily to the class of edgeless graphs).…”