20 research outputs found
AlonāTarsi Number and Modulo AlonāTarsi Number of Signed Graphs
Abstract(#br)We extend the concept of the AlonāTarsi number for unsigned graph to signed one. Moreover, we introduce the modulo AlonāTarsi number for a prime number p . We show that both the AlonāTarsi number and modulo AlonāTarsi number of a signed planar graph ( G , Ļ ) are at most 5, where the former result generalizes Zhuās result for unsigned case and the latter one implies that ( G , Ļ ) is Z 5 -colorable
Extensions of Galvin's Theorem
We discuss problems in list coloring with an emphasis on techniques that utilize oriented graphs. Our central theme is Galvin's resolution of the Dinitz problem (Galvin. J. Comb. Theory, Ser. B 63(1), 1995, 153--158).
We survey the related work of Alon and Tarsi (Combinatorica 12(2) 1992, 125--134) and H\"{a}ggkvist and Janssen (Combinatorics, Probability \& Computing 6(3) 1997,
295--313). We then prove two new extensions of Galvin's theorem
Distance-two coloring of sparse graphs
Consider a graph and, for each vertex , a subset
of neighbors of . A -coloring is a coloring of the
elements of so that vertices appearing together in some receive
pairwise distinct colors. An obvious lower bound for the minimum number of
colors in such a coloring is the maximum size of a set , denoted by
. In this paper we study graph classes for which there is a
function , such that for any graph and any , there is a
-coloring using at most colors. It is proved that if
such a function exists for a class , then can be taken to be a linear
function. It is also shown that such classes are precisely the classes having
bounded star chromatic number. We also investigate the list version and the
clique version of this problem, and relate the existence of functions bounding
those parameters to the recently introduced concepts of classes of bounded
expansion and nowhere-dense classes.Comment: 13 pages - revised versio
Coloring and constructing (hyper)graphs with restrictions
We consider questions regarding the existence of graphs and hypergraphs with certain coloring properties and other structural properties.
In Chapter 2 we consider color-critical graphs that are nearly bipartite and have few edges. We prove a conjecture of Chen, ErdÅs, GyĆ”rfĆ”s, and Schelp concerning the minimum number of edges in a ānearly bipartiteā 4-critical graph.
In Chapter 3 we consider coloring and list-coloring graphs and hypergraphs with few edges and no small cycles. We prove two main results. If a bipartite graph has maximum average degree at most 2(kā1), then it is colorable from lists of size k; we prove that this is sharp, even with an additional girth requirement. Using the same approach, we also provide a simple construction of graphs with arbitrarily large girth and chromatic number (first proved to exist by ErdÅs).
In Chapter 4 we consider list-coloring the family of kth power graphs. Kostochka and Woodall conjectured that graph squares are chromatic-choosable, as a strengthening of the Total List Coloring Conjecture. Kim and Park disproved this stronger conjecture, and Zhu asked whether graph kth powers are chromatic-choosable for any k. We show that this is not true: we construct families of graphs based on affine planes whose choice number exceeds their chromatic number by a logarithmic factor.
In Chapter 5 we consider the existence of uniform hypergraphs with prescribed degrees and codegrees. In Section 5.2, we show that a generalization of the graphic 2-switch is insufficient to connect realizations of a given degree sequence. In Section 5.3, we consider an operation on 3-graphs related to the octahedron that preserves codegrees; this leads to an inductive definition for 2-colorable triangulations of the sphere. In Section 5.4, we discuss the notion of fractional realizations of degree sequences, in particular noting the equivalence of the existence of a realization and the existence of a fractional realization in the graph and multihypergraph cases.
In Chapter 6 we consider a question concerning poset dimension. Dorais asked for the maximum guaranteed size of a subposet with dimension at most d of an n-element poset. A lower bound of sqrt(dn) was observed by Goodwillie. We provide a sublinear upper bound