2,078 research outputs found
Minimum Cost Homomorphisms to Locally Semicomplete and Quasi-Transitive Digraphs
For digraphs and , a homomorphism of to is a mapping $f:\
V(G)\dom V(H)uv\in A(G)f(u)f(v)\in A(H)u \in V(G)c_i(u), i \in V(H)f\sum_{u\in V(G)}c_{f(u)}(u)HHHGc_i(u)u\in V(G)i\in V(H)GH$ and, if one exists, to find one of minimum cost.
Minimum cost homomorphism problems encompass (or are related to) many well
studied optimization problems such as the minimum cost chromatic partition and
repair analysis problems. We focus on the minimum cost homomorphism problem for
locally semicomplete digraphs and quasi-transitive digraphs which are two
well-known generalizations of tournaments. Using graph-theoretic
characterization results for the two digraph classes, we obtain a full
dichotomy classification of the complexity of minimum cost homomorphism
problems for both classes
Graphs and graph polynomials
A dissertation submitted to the School of Mathematics in fulfilment of the requirements for the degree of Master of Science School of Mathematics University of the Witwatersrand, October 2017In this work we study the k-defect polynomials of a graph G. The k defect polynomial
is a function in λ that gives the number of improper colourings of a graph using
λ colours. The k-defect polynomials generate the bad colouring polynomial which
is equivalent to the Tutte polynomial, hence their importance in a more general
graph theoretic setting. By setting up a one-to-one correspondence between triangular
numbers and complete graphs, we use number theoretical methods to study certain
characteristics of the k-defect polynomials of complete graphs. Specifically we are able
to generate an expression for any k-defect polynomial of a complete graph, determine
integer intervals for k on which the k-defect polynomials for complete graphs are equal
to zero and also determine a formula to calculate the minimum number of k-defect
polynomials that are equal to zero for any complete graph.XL201
Minimum Cost Homomorphisms to Reflexive Digraphs
For digraphs and , a homomorphism of to is a mapping $f:\
V(G)\dom V(H)uv\in A(G)f(u)f(v)\in A(H)u \in V(G)c_i(u), i \in V(H)f\sum_{u\in V(G)}c_{f(u)}(u)H, the {\em minimum cost homomorphism problem} for HHGc_i(u)u\in V(G)i\in V(H)kGHk. We focus on the
minimum cost homomorphism problem for {\em reflexive} digraphs HHHH has a {\em Min-Max ordering}, i.e.,
if its vertices can be linearly ordered by <i<j, s<rir, js
\in A(H)is \in A(H)jr \in A(H)H$ which does not admit a Min-Max ordering, the minimum cost
homomorphism problem is NP-complete. Thus we obtain a full dichotomy
classification of the complexity of minimum cost homomorphism problems for
reflexive digraphs
Partition functions for states models on signed graphs
There are many models of physical systems based in unsigned graph theory, however there are some systems that cannot be modelled by unsigned graphs and that is the reason of signed graph states models. For these models, there are useful objects as partition functions, and systems that share these objects have some intrinsic properties in common, so it is interesting to build these models in order to relate those systems (with vertex switching invariance or equivalent knot invariance).Outgoin
Fast distributed approximation algorithms for vertex cover and set cover in anonymous networks
We present a distributed algorithm that finds a maximal edge packing in O(Δ + log* W) synchronous communication rounds in a weighted graph, independent of the number of nodes in the network; here Δ is the maximum degree of the graph and W is the maximum weight. As a direct application, we have a distributed 2-approximation algorithm for minimum-weight vertex cover, with the same running time. We also show how to find an f-approximation of minimum-weight set cover in O(f2k2 + fk log* W) rounds; here k is the maximum size of a subset in the set cover instance, f is the maximum frequency of an element, and W is the maximum weight of a subset. The algorithms are deterministic, and they can be applied in anonymous networks.Peer reviewe
List colouring hypergraphs and extremal results for acyclic graphs
We study several extremal problems in graphs and hypergraphs. The first one is on list-colouring hypergraphs, which is a generalization of the ordinary colouring of hypergraphs. We discuss two methods for determining the list-chromatic number of hypergraphs. One method uses hypergraph polynomials, which invokes Alon's combinatorial nullstellensatz. This method usually requires computer power to complete the calculations needed for even a modest-sized hypergraph. The other method is elementary, and uses the idea of minimum improper colourings. We apply these methods to various classes of hypergraphs, including the projective planes.
We focus on solving the list-colouring problem for Steiner triple systems (STS). It is not hard using either method to determine that Steiner triple systems of orders 7, 9 and 13 are 3-list-chromatic. For systems of order 15, we show that they are 4-list-colourable, but they are also ``almost'' 3-list-colourable. For all Steiner triple systems, we prove a couple of simple upper bounds on their list-chromatic numbers. Also, unlike ordinary colouring where a 3-chromatic STS exists for each admissible order, we prove using probabilistic methods that for every , every STS of high enough order is not -list-colourable.
The second problem is on embedding nearly-spanning bounded-degree trees in sparse graphs. We determine sufficient conditions based on expansion properties for a sparse graph to embed every nearly-spanning tree of bounded degree. We then apply this to random graphs, addressing a question of Alon, Krivelevich and Sudakov, and determine a probability where the random graph asymptotically almost surely contains every tree of bounded degree. This is nearly optimal in terms of the maximum degree of the trees that we embed.
Finally, we solve a problem that arises from quantum computing, which can be formulated as an extremal question about maximizing the size of a type of acyclic directed graph
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