100 research outputs found
Minimal reducible bounds for induced-hereditary properties
AbstractLet (Ma,⊆) and (La,⊆) be the lattices of additive induced-hereditary properties of graphs and additive hereditary properties of graphs, respectively. A property R∈Ma (∈La) is called a minimal reducible bound for a property P∈Ma (∈La) if in the interval (P,R) of the lattice Ma (La) there are only irreducible properties. The set of all minimal reducible bounds of a property P∈Ma in the lattice Ma we denote by BM(P). Analogously, the set of all minimal reducible bounds of a property P∈La in La is denoted by BL(P).We establish a method to determine minimal reducible bounds for additive degenerate induced-hereditary (hereditary) properties of graphs. We show that this method can be successfully used to determine already known minimal reducible bounds for k-degenerate graphs and outerplanar graphs in the lattice La. Moreover, in terms of this method we describe the sets of minimal reducible bounds for partial k-trees and the graphs with restricted order of components in La and k-degenerate graphs in Ma
Minimal reducible bounds for the class of k-degenerate graphs
AbstractLet (La,⊆) be the lattice of hereditary and additive properties of graphs. A reducible property R∈La is called minimal reducible bound for a property P∈La if in the interval (P,R) of the lattice La, there are only irreducible properties. We prove that the set B(Dk)={Dp∘Dq:k=p+q+1} is the covering set of minimal reducible bounds for the class Dk of all k-degenerate graphs
On percolation and the bunkbed conjecture
We study a problem on edge percolation on product graphs . Here
is any finite graph and consists of two vertices connected
by an edge. Every edge in is present with probability
independent of other edges. The Bunkbed conjecture states that for all and
the probability that is in the same component as is greater
than or equal to the probability that is in the same component as
for every pair of vertices .
We generalize this conjecture and formulate and prove similar statements for
randomly directed graphs. The methods lead to a proof of the original
conjecture for special classes of graphs , in particular outerplanar graphs.Comment: 13 pages, improved exposition thanks to anonymous referee. To appear
in CP
Online choosability of graphs
We study several problems in graph coloring. In list coloring, each vertex has a set of available colors and must be assigned a color from this set so that adjacent vertices receive distinct colors; such a coloring is an -coloring, and we then say that is -colorable. Given a graph and a function , we say that is -choosable if is -colorable for any list assignment such that for all . When for all and is -choosable, we say that is -choosable. The least such that is -choosable is the choice number, denoted . We focus on an online version of this problem, which is modeled by the Lister/Painter game.
The game is played on a graph in which every vertex has a positive number of tokens. In each round, Lister marks a nonempty subset of uncolored vertices, removing one token at each marked vertex. Painter responds by selecting a subset of that forms an independent set in . A color distinct from those used on previous rounds is given to all vertices in . Lister wins by marking a vertex that has no tokens, and Painter wins by coloring all vertices in . When Painter has a winning strategy, we say that is -paintable. If for all and is -paintable, then we say that is -paintable. The least such that is -paintable is the paint number, denoted \pa(G).
In Chapter 2, we develop useful tools for studying the Lister/Painter game. We study the paintability of graph joins and of complete bipartite graphs. In particular, \pa(K_{k,r})\le k if and only if .
In Chapter 3, we study the Lister/Painter game with the added restriction that the proper coloring produced by Painter must also satisfy some property . The main result of Chapter 3 provides a general method to give a winning strategy for Painter when a strategy for the list coloring problem is already known. One example of a property is that of having an -dynamic coloring, where a proper coloring is -dynamic if each vertex has at least distinct colors in its neighborhood. For any graph and any , we give upper bounds on how many tokens are necessary for Painter to produce an -dynamic coloring of . The upper bounds are in terms of and the genus of a surface on which embeds.
In Chapter 4, we study a version of the Lister/Painter game in which Painter must assign colors to each vertex so that adjacent vertices receive disjoint color sets. We characterize the graphs in which tokens is sufficient to produce such a coloring. We strengthen Brooks' Theorem as well as Thomassen's result that planar graphs are 5-choosable.
In Chapter 5, we study sum-paintability. The sum-paint number of a graph , denoted \spa(G), is the least over all such that is -paintable. We prove the easy upper bound: \spa(G)\le|V(G)|+|E(G)|. When \spa(G)=|V(G)|+|E(G)|, we say that is sp-greedy. We determine the sum-paintability of generalized theta-graphs. The generalized theta-graph consists of two vertices joined by paths of lengths \VEC \ell1k. We conjecture that outerplanar graphs are sp-greedy and prove several partial results toward this conjecture.
In Chapter 6, we study what happens when Painter is allowed to allocate tokens as Lister marks vertices. The slow-coloring game is played by Lister and Painter on a graph . Lister marks a nonempty set of uncolored vertices and scores 1 point for each marked vertex. Painter colors all vertices in an independent subset of the marked vertices with a color distinct from those used previously in the game. The game ends when all vertices have been colored. The sum-color cost of a graph , denoted \scc(G), is the maximum score Lister can guarantee in the slow-coloring game on . We prove several general lower and upper bounds for \scc(G). In more detail, we study trees and prove sharp upper and lower bounds over all trees with vertices. We give a formula to determine \scc(G) exactly when . Separately, we prove that \scc(G)=\spa(G) if and only if is a disjoint union of cliques. Lastly, we give lower and upper bounds on \scc(K_{r,s})
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