38 research outputs found
Universal targets for homomorphisms of edge-colored graphs
A -edge-colored graph is a finite, simple graph with edges labeled by
numbers . A function from the vertex set of one -edge-colored
graph to another is a homomorphism if the endpoints of any edge are mapped to
two different vertices connected by an edge of the same color. Given a class
of graphs, a -edge-colored graph (not necessarily
with the underlying graph in ) is -universal for
when any -edge-colored graph with the underlying graph in
admits a homomorphism to . We characterize graph classes that admit
-universal graphs. For such classes, we establish asymptotically almost
tight bounds on the size of the smallest universal graph.
For a nonempty graph , the density of is the maximum ratio of the
number of edges to the number of vertices ranging over all nonempty subgraphs
of . For a nonempty class of graphs, denotes
the density of , that is the supremum of densities of graphs in
.
The main results are the following. The class admits
-universal graphs for if and only if there is an absolute constant
that bounds the acyclic chromatic number of any graph in . For any
such class, there exists a constant , such that for any , the size
of the smallest -universal graph is between and
.
A connection between the acyclic coloring and the existence of universal
graphs was first observed by Alon and Marshall (Journal of Algebraic
Combinatorics, 8(1):5-13, 1998). One of their results is that for planar
graphs, the size of the smallest -universal graph is between and
. Our results yield that there exists a constant such that for all
, this size is bounded from above by
Chip games and paintability
We prove that the difference between the paint number and the choice number
of a complete bipartite graph is . That answers
the question of Zhu (2009) whether this difference, for all graphs, can be
bounded by a common constant. By a classical correspondence, our result
translates to the framework of on-line coloring of uniform hypergraphs. This
way we obtain that for every on-line two coloring algorithm there exists a
k-uniform hypergraph with edges on which the strategy fails. The
results are derived through an analysis of a natural family of chip games
Defective 3-Paintability of Planar Graphs
A -defective -painting game on a graph is played by two players:
Lister and Painter. Initially, each vertex is uncolored and has tokens. In
each round, Lister marks a chosen set of uncolored vertices and removes one
token from each marked vertex. In response, Painter colors vertices in a subset
of which induce a subgraph of maximum degree at most . Lister
wins the game if at the end of some round there is an uncolored vertex that has
no more tokens left. Otherwise, all vertices eventually get colored and Painter
wins the game. We say that is -defective -paintable if Painter has a
winning strategy in this game. In this paper we show that every planar graph is
3-defective 3-paintable and give a construction of a planar graph that is not
2-defective 3-paintable.Comment: 21 pages, 11 figure
A Note on Two-Colorability of Nonuniform Hypergraphs
For a hypergraph , let denote the expected number of monochromatic
edges when the color of each vertex in is sampled uniformly at random from
the set of size 2. Let denote the minimum size of an edge in .
Erd\H{o}s asked in 1963 whether there exists an unbounded function such
that any hypergraph with and is two
colorable. Beck in 1978 answered this question in the affirmative for a
function . We improve this result by showing that, for
an absolute constant , a version of random greedy coloring procedure
is likely to find a proper two coloring for any hypergraph with
and
Graph Polynomials and Group Coloring of Graphs
Let be an Abelian group and let be a simple graph. We say that
is -colorable if for some fixed orientation of and every edge
labeling , there exists a vertex coloring by
the elements of such that , for every edge
(oriented from to ).
Langhede and Thomassen proved recently that every planar graph on
vertices has at least different -colorings. By using a
different approach based on graph polynomials, we extend this result to
-minor-free graphs in the more general setting of field coloring. More
specifically, we prove that every such graph on vertices is
--choosable, whenever is an arbitrary field with at
least elements. Moreover, the number of colorings (for every list
assignment) is at least .Comment: 14 page
Online coloring of short intervals
We study the online graph coloring problem restricted to the intersection graphs of intervals withlengths in[1,σ]. Forσ= 1it is the class of unit interval graphs, and forσ=∞the class of allinterval graphs. Our focus is on intermediary classes. We present a(1 +σ)-competitive algorithm,which beats the state of the art for11, nor better than7/4-competitive for anyσ >2, and that no algorithm beats the5/2asymptotic competitive ratio for all, arbitrarily large,values ofσ. That last result shows that the problem we study can be strictly harder than unitinterval coloring. Our main technical contribution is a recursive composition of strategies, whichseems essential to prove any lower bound higher than2
Coloring and Recognizing Directed Interval Graphs
A \emph{mixed interval graph} is an interval graph that has, for every pair
of intersecting intervals, either an arc (directed arbitrarily) or an
(undirected) edge. We are particularly interested in scenarios where edges and
arcs are defined by the geometry of intervals. In a proper coloring of a mixed
interval graph , an interval receives a lower (different) color than an
interval if contains arc (edge ). Coloring of mixed
graphs has applications, for example, in scheduling with precedence
constraints; see a survey by Sotskov [Mathematics, 2020]. For coloring general
mixed interval graphs, we present a -approximation algorithm, where is the size of a largest clique
and is the length of a longest directed path in . For the
subclass of \emph{bidirectional interval graphs} (introduced recently for an
application in graph drawing), we show that optimal coloring is NP-hard. This
was known for general mixed interval graphs. We introduce a new natural class
of mixed interval graphs, which we call \emph{containment interval graphs}. In
such a graph, there is an arc if interval contains interval ,
and there is an edge if and overlap. We show that these
graphs can be recognized in polynomial time, that coloring them with the
minimum number of colors is NP-hard, and that there is a 2-approximation
algorithm for coloring.Comment: To appear in Proc. ISAAC 202