14,104 research outputs found
Oriented coloring on recursively defined digraphs
Coloring is one of the most famous problems in graph theory. The coloring
problem on undirected graphs has been well studied, whereas there are very few
results for coloring problems on directed graphs. An oriented k-coloring of an
oriented graph G=(V,A) is a partition of the vertex set V into k independent
sets such that all the arcs linking two of these subsets have the same
direction. The oriented chromatic number of an oriented graph G is the smallest
k such that G allows an oriented k-coloring. Deciding whether an acyclic
digraph allows an oriented 4-coloring is NP-hard. It follows, that finding the
chromatic number of an oriented graph is an NP-hard problem. This motivates to
consider the problem on oriented co-graphs. After giving several
characterizations for this graph class, we show a linear time algorithm which
computes an optimal oriented coloring for an oriented co-graph. We further
prove how the oriented chromatic number can be computed for the disjoint union
and order composition from the oriented chromatic number of the involved
oriented co-graphs. It turns out that within oriented co-graphs the oriented
chromatic number is equal to the length of a longest oriented path plus one. We
also show that the graph isomorphism problem on oriented co-graphs can be
solved in linear time.Comment: 14 page
List Distinguishing Parameters of Trees
A coloring of the vertices of a graph G is said to be distinguishing}
provided no nontrivial automorphism of G preserves all of the vertex colors.
The distinguishing number of G, D(G), is the minimum number of colors in a
distinguishing coloring of G. The distinguishing chromatic number of G,
chi_D(G), is the minimum number of colors in a distinguishing coloring of G
that is also a proper coloring.
Recently the notion of a distinguishing coloring was extended to that of a
list distinguishing coloring. Given an assignment L= {L(v) : v in V(G)} of
lists of available colors to the vertices of G, we say that G is (properly)
L-distinguishable if there is a (proper) distinguishing coloring f of G such
that f(v) is in L(v) for all v. The list distinguishing number of G, D_l(G), is
the minimum integer k such that G is L-distinguishable for any list assignment
L with |L(v)| = k for all v. Similarly, the list distinguishing chromatic
number of G, denoted chi_{D_l}(G) is the minimum integer k such that G is
properly L-distinguishable for any list assignment L with |L(v)| = k for all v.
In this paper, we study these distinguishing parameters for trees, and in
particular extend an enumerative technique of Cheng to show that for any tree
T, D_l(T) = D(T), chi_D(T)=chi_{D_l}(T), and chi_D(T) <= D(T) + 1.Comment: 10 page
Vertex arboricity of triangle-free graphs
Master's Project (M.S.) University of Alaska Fairbanks, 2016The vertex arboricity of a graph is the minimum number of colors needed to color the vertices so that the subgraph induced by each color class is a forest. In other words, the vertex arboricity of a graph is the fewest number of colors required in order to color a graph such that every cycle has at least two colors. Although not standard, we will refer to vertex arboricity simply as arboricity. In this paper, we discuss properties of chromatic number and k-defective chromatic number and how those properties relate to the arboricity of trianglefree graphs. In particular, we find bounds on the minimum order of a graph having arboricity three. Equivalently, we consider the largest possible vertex arboricity of triangle-free graphs of fixed order
A Survey on Monochromatic Connections of Graphs
The concept of monochromatic connection of graphs was introduced by Caro and
Yuster in 2011. Recently, a lot of results have been published about it. In
this survey, we attempt to bring together all the results that dealt with it.
We begin with an introduction, and then classify the results into the following
categories: monochromatic connection coloring of edge-version, monochromatic
connection coloring of vertex-version, monochromatic index, monochromatic
connection coloring of total-version.Comment: 26 pages, 3 figure
Algorithms and almost tight results for 3-colorability of small diameter graphs.
The 3-coloring problem is well known to be NP-complete. It is also well known that it remains NP-complete when the input is restricted to graphs with diameter 4. Moreover, assuming the Exponential Time Hypothesis (ETH), 3-coloring cannot be solved in time 2o(n) on graphs with n vertices and diameter at most 4. In spite of extensive studies of the 3-coloring problem with respect to several basic parameters, the complexity status of this problem on graphs with small diameter, i.e. with diameter at most 2, or at most 3, has been an open problem. In this paper we investigate graphs with small diameter. For graphs with diameter at most 2, we provide the first subexponential algorithm for 3-coloring, with complexity 2O(nlogn√). Furthermore we extend the notion of an articulation vertex to that of an articulation neighborhood, and we provide a polynomial algorithm for 3-coloring on graphs with diameter 2 that have at least one articulation neighborhood. For graphs with diameter at most 3, we establish the complexity of 3-coloring by proving for every ε∈[0,1) that 3-coloring is NP-complete on triangle-free graphs of diameter 3 and radius 2 with n vertices and minimum degree δ=Θ(nε). Moreover, assuming ETH, we use three different amplification techniques of our hardness results, in order to obtain for every ε∈[0,1) subexponential asymptotic lower bounds for the complexity of 3-coloring on triangle-free graphs with diameter 3 and minimum degree δ=Θ(nε). Finally, we provide a 3-coloring algorithm with running time 2O(min{δΔ, nδlogδ}) for arbitrary graphs with diameter 3, where n is the number of vertices and δ (resp. Δ) is the minimum (resp. maximum) degree of the input graph. To the best of our knowledge, this is the first subexponential algorithm for graphs with δ=ω(1) and for graphs with δ=O(1) and Δ=o(n). Due to the above lower bounds of the complexity of 3-coloring, the running time of this algorithm is asymptotically almost tight when the minimum degree of the input graph is δ=Θ(nε), where ε∈[12,1), as its time complexity is 2O(nδlogδ)=2O(n1−εlogn) and the corresponding lower bound states that there is no 2o(n1−ε)-time algorithm
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