Sum-Paintability of Generalized Theta-Graphs

Abstract

In online list coloring (introduced by Zhu and by Schauz in 2009), on each round the set of vertices having a particular color in their lists is revealed, and the coloring algorithm chooses an independent subset of this set to receive that color. For a graph G and a function f : V (G) → N, the graph is f -paintable if there is an algorithm to produce a proper coloring when each vertex v is allowed to be presented at most f (v) times.The sum-paintability. . , H k are the blocks of G. Also, adding an ear of length ℓ to G adds 2ℓ − 1 to the sum-paintability, when ℓ ≥ 3.Strengthening a result of Berliner et al., we prove χ sp (K 2,r ) = 2r+min{l+m : lm > r}. The generalized theta-graph Θ ℓ 1 ,...,ℓ k consists of two vertices joined by internally disjoint paths of lengths ℓ 1 , . . . , ℓ k . A book is a graph of the form Θ 1,2,...,2 , denoted B r when there are r internally disjoint paths of length 2. We prove χ sp (B r ) = 2r + min l,m∈N {l + m : m(l − m) + m 2 > r}. We use these results to determine the sumpaintability for all generalized theta-graphs.

Strong edge geodetic problem in networks

“…To show that the bound of Theorem 6.2 is sharp, let us say that a graph G is r-good if it contains vertices u and v such that I.u; v/ contains r shortest u; v-paths which cover all the edges of G. For example, uniform theta graphs [20] are r-good graphs. To specify the vertices u and v we will denote such a graph with G Proof.…”

Families of Online Sum-Choice-Greedy Graphs

“…Note that this necessary condition for G to be sp-greedy is not sufficient, since Carraher et al [3] showed that χ sp ( 2,2,2 ) = 10 < σ( 2,2,2 ) = 11. Thus 2,2,2 is not spgreedy, but deleting a vertex leaves K 1,3 or C 4 , both of which are sp-greedy.…”

“…Other planar graphs, however, are not spgreedy. Carraher et al [3] showed that not all generalized theta-graphs are sp-greedy; for example, any planar graph containing K 2 K 4 is not sp-greedy. Though K 2 K 4 has embeddings for which the weak dual is a multigraph with a path as the underlying graph, it is not outerplanar.…”

The interactive sum choice number of graphs

“…Two other variants of sum-choosability have also been introduced recently. In the sumpaintability variant [3,14], the painter decides a budget for each vertex in advance (as in sum list colouring), then in each round the lister reveals a subset of vertices which have colour c in the list and the painter must decide immediately which of these vertices to paint with colour c. Thus, there is less information available than in the standard setting of sum-choosability since painter must fix the colour of some vertices before knowing the entire contents of the colour lists. The relationship between the interactive sum choice number and the second of these variants, the slow-colouring game [13,15] is less clear.…”