1,027 research outputs found
Packing Coloring of Undirected and Oriented Generalized Theta Graphs
The packing chromatic number (G) of an undirected (resp.
oriented) graph G is the smallest integer k such that its set of vertices V (G)
can be partitioned into k disjoint subsets V 1,..., V k, in such a way that
every two distinct vertices in V i are at distance (resp. directed distance)
greater than i in G for every i, 1 i k. The generalized theta graph
{\ell} 1,...,{\ell}p consists in two end-vertices joined by p 2
internally vertex-disjoint paths with respective lengths 1 {\ell} 1
. . . {\ell} p. We prove that the packing chromatic number of any
undirected generalized theta graph lies between 3 and max{5, n 3 + 2}, where n
3 = |{i / 1 i p, {\ell} i = 3}|, and that both these bounds are
tight. We then characterize undirected generalized theta graphs with packing
chromatic number k for every k 3. We also prove that the packing
chromatic number of any oriented generalized theta graph lies between 2 and 5
and that both these bounds are tight.Comment: Revised version. Accepted for publication in Australas. J. Combi
Backbone colorings for networks: tree and path backbones
We introduce and study backbone colorings, a variation on classical vertex colorings: Given a graph and a spanning subgraph of (the backbone of ), a backbone coloring for and is a proper vertex coloring of in which the colors assigned to adjacent vertices in differ by at least two. We study the cases where the backbone is either a spanning tree or a spanning path
Distinguishing locally finite trees
The distinguishing number of a graph is the smallest number of
colors that is needed to color the vertices of such that the only color
preserving automorphism is the identity. For infinite graphs is bounded
by the supremum of the valences, and for finite graphs by , where
is the maximum valence. Given a finite or infinite tree of
bounded finite valence and an integer , where , we are
interested in coloring the vertices of by colors, such that every color
preserving automorphism fixes as many vertices as possible. In this sense we
show that there always exists a -coloring for which all vertices whose
distance from the next leaf is at least are fixed by any
color preserving automorphism, and that one can do much better in many cases
On weight distributions of perfect colorings and completely regular codes
A vertex coloring of a graph is called "perfect" if for any two colors
and , the number of the color- neighbors of a color- vertex does
not depend on the choice of , that is, depends only on and (the
corresponding partition of the vertex set is known as "equitable"). A set of
vertices is called "completely regular" if the coloring according to the
distance from this set is perfect. By the "weight distribution" of some
coloring with respect to some set we mean the information about the number of
vertices of every color at every distance from the set. We study the weight
distribution of a perfect coloring (equitable partition) of a graph with
respect to a completely regular set (in particular, with respect to a vertex if
the graph is distance-regular). We show how to compute this distribution by the
knowledge of the color composition over the set. For some partial cases of
completely regular sets, we derive explicit formulas of weight distributions.
Since any (other) completely regular set itself generates a perfect coloring,
this gives universal formulas for calculating the weight distribution of any
completely regular set from its parameters. In the case of Hamming graphs, we
prove a very simple formula for the weight enumerator of an arbitrary perfect
coloring. Codewords: completely regular code; equitable partition; partition
design; perfect coloring; perfect structure; regular partition; weight
distribution; weight enumerator.Comment: 17pp; partially presented at "Optimal Codes and Related Topics"
OC2009, Varna (Bulgaria). V.2: the title was changed (old: "On weight
distributions of perfect structures"), Sect.5 "Weight enumerators ..." was
adde
Totally Silver Graphs
A totally silver coloring of a graph G is a k--coloring of G such that for
every vertex v \in V(G), each color appears exactly once on N[v], the closed
neighborhood of v. A totally silver graph is a graph which admits a totally
silver coloring. Totally silver coloring are directly related to other areas of
graph theory such as distance coloring and domination. In this work, we present
several constructive characterizations of totally silver graphs and bipartite
totally silver graphs. We give several infinite families of totally silver
graphs. We also give cubic totally silver graphs of girth up to 10
The chromatic number of the plane is at least 5 - a new proof
We present an alternate proof of the fact that given any 4-coloring of the
plane there exist two points unit distance apart which are identically colored.Comment: 12 pages, 7 figure
Conflict-Free Coloring of Planar Graphs
A conflict-free k-coloring of a graph assigns one of k different colors to
some of the vertices such that, for every vertex v, there is a color that is
assigned to exactly one vertex among v and v's neighbors. Such colorings have
applications in wireless networking, robotics, and geometry, and are
well-studied in graph theory. Here we study the natural problem of the
conflict-free chromatic number chi_CF(G) (the smallest k for which
conflict-free k-colorings exist). We provide results both for closed
neighborhoods N[v], for which a vertex v is a member of its neighborhood, and
for open neighborhoods N(v), for which vertex v is not a member of its
neighborhood.
For closed neighborhoods, we prove the conflict-free variant of the famous
Hadwiger Conjecture: If an arbitrary graph G does not contain K_{k+1} as a
minor, then chi_CF(G) <= k. For planar graphs, we obtain a tight worst-case
bound: three colors are sometimes necessary and always sufficient. We also give
a complete characterization of the computational complexity of conflict-free
coloring. Deciding whether chi_CF(G)<= 1 is NP-complete for planar graphs G,
but polynomial for outerplanar graphs. Furthermore, deciding whether
chi_CF(G)<= 2 is NP-complete for planar graphs G, but always true for
outerplanar graphs. For the bicriteria problem of minimizing the number of
colored vertices subject to a given bound k on the number of colors, we give a
full algorithmic characterization in terms of complexity and approximation for
outerplanar and planar graphs.
For open neighborhoods, we show that every planar bipartite graph has a
conflict-free coloring with at most four colors; on the other hand, we prove
that for k in {1,2,3}, it is NP-complete to decide whether a planar bipartite
graph has a conflict-free k-coloring. Moreover, we establish that any general}
planar graph has a conflict-free coloring with at most eight colors.Comment: 30 pages, 17 figures; full version (to appear in SIAM Journal on
Discrete Mathematics) of extended abstract that appears in Proceeedings of
the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA
2017), pp. 1951-196
List-coloring embedded graphs
For any fixed surface Sigma of genus g, we give an algorithm to decide
whether a graph G of girth at least five embedded in Sigma is colorable from an
assignment of lists of size three in time O(|V(G)|). Furthermore, we can allow
a subgraph (of any size) with at most s components to be precolored, at the
expense of increasing the time complexity of the algorithm to
O(|V(G)|^{K(g+s)+1}) for some absolute constant K; in both cases, the
multiplicative constant hidden in the O-notation depends on g and s. This also
enables us to find such a coloring when it exists. The idea of the algorithm
can be applied to other similar problems, e.g., 5-list-coloring of graphs on
surfaces.Comment: 14 pages, 0 figures, accepted to SODA'1
Star chromatic index
The star chromatic index of a graph is the minimum number of
colors needed to properly color the edges of the graph so that no path or cycle
of length four is bi-colored. We obtain a near-linear upper bound in terms of
the maximum degree . Our best lower bound on in
terms of is valid for complete graphs. We also
consider the special case of cubic graphs, for which we show that the star
chromatic index lies between 4 and 7 and characterize the graphs attaining the
lower bound. The proofs involve a variety of notions from other branches of
mathematics and may therefore be of certain independent interest.Comment: 16 pages, 3 figure
The complexity of nonrepetitive edge coloring of graphs
A squarefree word is a sequence of symbols such that there are no strings
, and for which . A nonrepetitive coloring of a graph is an
edge coloring in which the sequence of colors along any open path is
squarefree. We show that determining whether a graph has a nonrepetitive
-coloring is -complete. When we restrict to paths of lengths at
most , the problem becomes NP-complete for fixed
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