3,579 research outputs found
Hereditary Graph Classes: When the Complexities of Colouring and Clique Cover Coincide
A graph is -free for a pair of graphs if it contains no
induced subgraph isomorphic to or . In 2001, Kr\'al',
Kratochv\'{\i}l, Tuza, and Woeginger initiated a study into the complexity of
Colouring for -free graphs. Since then, others have tried to
complete their study, but many cases remain open. We focus on those
-free graphs where is , the complement of
. As these classes are closed under complementation, the computational
complexities of Colouring and Clique Cover coincide. By combining new and known
results, we are able to classify the complexity of Colouring and Clique Cover
for -free graphs for all cases except when for
or for . We also classify the complexity of
Colouring on graph classes characterized by forbidding a finite number of
self-complementary induced subgraphs, and we initiate a study of -Colouring
for -free graphs.Comment: 19 Pages, 5 Figure
Topological Graphic Passwords And Their Matchings Towards Cryptography
Graphical passwords (GPWs) are convenient for mobile equipments with touch
screen. Topological graphic passwords (Topsnut-gpws) can be saved in computer
by classical matrices and run quickly than the existing GPWs. We research
Topsnut-gpws by the matching of view, since they have many advantages. We
discuss: configuration matching partition, coloring/labelling matching
partition, set matching partition, matching chain, etc. And, we introduce new
graph labellings for enriching Topsnut-matchings and show that these labellings
can be realized for trees or spanning trees of networks. In theoretical works
we explore Graph Labelling Analysis, and show that every graph admits our
extremal labellings and set-type labellings in graph theory. Many of the graph
labellings mentioned are related with problems of set matching partitions to
number theory, and yield new objects and new problems to graph theory
Text-based Passwords Generated From Topological Graphic Passwords
Topological graphic passwords (Topsnut-gpws) are one of graph-type passwords,
but differ from the existing graphical passwords, since Topsnut-gpws are saved
in computer by algebraic matrices. We focus on the transformation between
text-based passwords (TB-paws) and Topsnut-gpws in this article. Several
methods for generating TB-paws from Topsnut-gpws are introduced; these methods
are based on topological structures and graph coloring/labellings, such that
authentications must have two steps: one is topological structure
authentication, and another is text-based authentication. Four basic
topological structure authentications are introduced and many text-based
authentications follow Topsnut-gpws. Our methods are based on algebraic, number
theory and graph theory, many of them can be transformed into polynomial
algorithms. A new type of matrices for describing Topsnut-gpws is created here,
and such matrices can produce TB-paws in complex forms and longer bytes.
Estimating the space of TB-paws made by Topsnut-gpws is very important for
application. We propose to encrypt dynamic networks and try to face: (1)
thousands of nodes and links of dynamic networks; (2) large numbers of
Topsnut-gpws generated by machines rather than human's hands. As a try, we
apply spanning trees of dynamic networks and graphic groups (Topsnut-groups) to
approximate the solutions of these two problems. We present some unknown
problems in the end of the article for further research
Homomorphisms of signed planar graphs
Signed graphs are studied since the middle of the last century. Recently, the
notion of homomorphism of signed graphs has been introduced since this notion
captures a number of well known conjectures which can be reformulated using the
definitions of signed homomorphism.
In this paper, we introduce and study the properties of some target graphs
for signed homomorphism. Using these properties, we obtain upper bounds on the
signed chromatic numbers of graphs with bounded acyclic chromatic number and of
signed planar graphs with given girth
Unique colorability and clique minors
For a graph G, let h(G) denote the largest k such that G has k pairwise
disjoint pairwise adjacent connected nonempty subgraphs, and let s(G) denote
the largest k such that G has k pairwise disjoint pairwise adjacent connected
subgraphs of size 1 or 2. Hadwiger's conjecture states that h(G) is at least
c(G), where c(G) is the chromatic number of G. Seymour conjectured that s(G) is
at least |V(G)|/2 for all graphs without antitriangles, i. e. three pairwise
nonadjacent vertices. Here we concentrate on graphs G with exactly one
c(G)-coloring. We prove generalizations of (i) if c(G) is at most 6 and G has
exactly one c(G)-coloring then h(G) is at least c(G), where the proof does not
use the four-color-theorem, and (ii) if G has no antitriangles and G has
exactly one c(G)-coloring then s(G) is at least |V(G)|/2
Sigma-Adequate Link Diagrams and the Tutte Polynomial
In this paper, we characterize the sigma-adequacy of a link diagram in two
ways: in terms of a certain edge subset of its Tait graph and in terms of a
certain product of Tutte polynomials. Furthermore, we show that the symmetrized
Tutte polynomial of the Tait graph of a link diagram can be written as a sum of
these products of Tutte polynomials, where the sum is over the sigma-adequate
states of the given link diagram. Using this state sum, we show that the number
of sigma-adequate states of a link diagram is bounded above by the number of
spanning trees in its associated Tait graph. By combining results, we give a
method to find all of the sigma-adequate states of a link diagram. Finally, we
give necessary and sufficient conditions for a link diagram to be
sigma-adequate and sigma-homogeneous (also called homogeneously adequate) with
respect to a given state.Comment: 31 pages, 11 figure
On colorful edge triples in edge-colored complete graphs
An edge-coloring of the complete graph we call -caring if it leaves
no -subgraph of monochromatic and at the same time every subset of
vertices contains in it at least one completely multicolored version
of . For the first two meaningful cases, when and we
determine for infinitely many the minimum number of colors needed for an
-caring edge-coloring of . An explicit family of -edge-colorings of so that every quadruple of its vertices
contains a totally multicolored in at least one of them is also
presented. Investigating related Ramsey-type problems we also show that the
Shannon (OR-)capacity of the Gr\"otzsch graph is strictly larger than that of
the five length cycle.Comment: 15 page
A generalization of Witsenhausen's zero-error rate for directed graphs
We investigate a communication setup where a source output is sent through a
free noisy channel first and an additional codeword is sent through a noiseless
but expensive channel later. With the help of the second message the decoder
should be able to decide with zero-error whether its decoding of the first
message was error-free. This scenario leads to the definition of a digraph
parameter that generalizes Witsenhausen's zero-error rate for directed graphs.
We investigate this new parameter for some specific directed graphs and explore
its relations to other digraph parameters like Sperner capacity and dichromatic
number.
When the original problem is modified to require zero-error decoding of the
whole message then we arrive back to the Witsenhausen rate of an appropriately
defined undirected graph.Comment: 27 pages, 4 figure
Coloring (, bull)-free graphs
We give a polynomial-time algorithm that computes the chromatic number of any
graph that contains no path on five vertices and no bull as an induced subgraph
(where the bull is the graph with five vertices and edges
).Comment: 8 page
Regarding two conjectures on clique and biclique partitions
For a graph , let denote the minimum number of cliques of
needed to cover the edges of exactly once. Similarly, let denote
the minimum number of bicliques (i.e. complete bipartite subgraphs of )
needed to cover each edge of exactly times. We consider two conjectures
-- one regarding the maximum possible value of (due
to de Caen, Erd\H{o}s, Pullman and Wormald) and the other regarding
(due to de Caen, Gregory and Pritikin). We disprove the first, obtaining
improved lower and upper bounds on , and we
prove an asymptotic version of the second, showing that
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