221,490 research outputs found
Relaxation of Planar Graphs With d∆≥2 and No 4-Cycles
First, let a graph be a set of vertices (points) and a set of edges (lines) connecting these vertices. Further, let a planar graph be a graph that can be represented on the plane without any edges crossing. Define a (c1, c2,…ck)-coloring of graph G from G to {1,2,…k} such that for every i, 1≤ i ≤ k, G[Vi] has maximum degree at most ci, where G[Vi] represents the subgraph induced by the vertices colored with i. The Four Color Theorem by Appel and Haken (1973) states that all planar graphs are 4-colorable. The Bordeaux Conjecture (2003) postulates that planar graphs with no 5-cycles and without intersecting triangles is 3-colorable. Liu, Li and Yu (2014) proved that planar graphs without intersecting triangles and 5-cycles is (2,0,0) colorable. We prove that all planar graphs without 4-cycles and no less than two edges between triangles is also (2,0,0) colorable
A Class of Quantum LDPC Codes Constructed From Finite Geometries
Low-density parity check (LDPC) codes are a significant class of classical
codes with many applications. Several good LDPC codes have been constructed
using random, algebraic, and finite geometries approaches, with containing
cycles of length at least six in their Tanner graphs. However, it is impossible
to design a self-orthogonal parity check matrix of an LDPC code without
introducing cycles of length four.
In this paper, a new class of quantum LDPC codes based on lines and points of
finite geometries is constructed. The parity check matrices of these codes are
adapted to be self-orthogonal with containing only one cycle of length four.
Also, the column and row weights, and bounds on the minimum distance of these
codes are given. As a consequence, the encoding and decoding algorithms of
these codes as well as their performance over various quantum depolarizing
channels will be investigated.Comment: 5pages, 2 figure
Coloring the Square of Planar Graphs Without 4-Cycles or 5-Cycles
The famous Four Color Theorem states that any planar graph can be properly colored using at most four colors. However, if we want to properly color the square of a planar graph (or alternatively, color the graph using distinct colors on vertices at distance up to two from each other), we will always require at least \Delta + 1 colors, where \Delta is the maximum degree in the graph. For all \Delta, Wegner constructed planar graphs (even without 3-cycles) that require about \frac{3}{2} \Delta colors for such a coloring.
To prove a stronger upper bound, we consider only planar graphs that contain no 4-cycles and no 5-cycles (but which may contain 3-cycles). Zhu, Lu, Wang, and Chen showed that for a graph G in this class with \Delta \ge 9, we can color G^2 using no more than \Delta + 5 colors. In this thesis we improve this result, showing that for a planar graph G with maximum degree \Delta \ge 32 having no 4-cycles and no 5-cycles, at most \Delta + 3 colors are needed to properly color G^2. Our approach uses the discharging method, and the result extends to list-coloring and other related coloring concepts as well
Acyclic edge coloring of graphs
An {\em acyclic edge coloring} of a graph is a proper edge coloring such
that the subgraph induced by any two color classes is a linear forest (an
acyclic graph with maximum degree at most two). The {\em acyclic chromatic
index} \chiup_{a}'(G) of a graph is the least number of colors needed in
an acyclic edge coloring of . Fiam\v{c}\'{i}k (1978) conjectured that
\chiup_{a}'(G) \leq \Delta(G) + 2, where is the maximum degree of
. This conjecture is well known as Acyclic Edge Coloring Conjecture (AECC).
A graph with maximum degree at most is {\em
-deletion-minimal} if \chiup_{a}'(G) > \kappa and \chiup_{a}'(H)
\leq \kappa for every proper subgraph of . The purpose of this paper is
to provide many structural lemmas on -deletion-minimal graphs. By using
the structural lemmas, we firstly prove that AECC is true for the graphs with
maximum average degree less than four (\autoref{NMAD4}). We secondly prove that
AECC is true for the planar graphs without triangles adjacent to cycles of
length at most four, with an additional condition that every -cycle has at
most three edges contained in triangles (\autoref{NoAdjacent}), from which we
can conclude some known results as corollaries. We thirdly prove that every
planar graph without intersecting triangles satisfies \chiup_{a}'(G) \leq
\Delta(G) + 3 (\autoref{NoIntersect}). Finally, we consider one extreme case
and prove it: if is a graph with and all the
-vertices are independent, then \chiup_{a}'(G) = \Delta(G). We hope
the structural lemmas will shed some light on the acyclic edge coloring
problems.Comment: 19 page
Hamiltonian cycles in maximal planar graphs and planar triangulations
In this thesis we study planar graphs, in particular, maximal planar graphs and general planar triangulations. In Chapter 1 we present the terminology and notations that will be used throughout the thesis and review some elementary results on graphs that we shall need. In Chapter 2 we study the fundamentals of planarity, since it is the cornerstone of this thesis. We begin with the famous Euler's Formula which will be used in many of our results. Then we discuss another famous theorem in graph theory, the Four Colour Theorem. Lastly, we discuss Kuratowski's Theorem, which gives a characterization of planar graphs. In Chapter 3 we discuss general properties of a maximal planar graph, G particularly concerning connectivity. First we discuss maximal planar graphs with minimum degree i, for i = 3; 4; 5, and the subgraph induced by the vertices of G with the same degree. Finally we discuss the connectivity of G, a maximal planar graph with minimum degree i. Chapter 4 will be devoted to Hamiltonian cycles in maximal planar graphs. We discuss the existence of Hamiltonian cycles in maximal planar graphs. Whitney proved that any maximal planar graph without a separating triangle is Hamiltonian, where a separating triangle is a triangle such that its removal disconnects the graph. Chen then extended Whitney's results and allowed for one separating triangle and showed that the graph is still Hamiltonian. Helden also extended Chen's result and allowed for two separating triangles and showed that the graph is still Hamiltonian. G. Helden and O. Vieten went further and allowed for three separating triangles and showed that the graph is still Hamiltonian. In the second section we discuss the question by Hakimi and Schmeichel: what is the number of cycles of length p that a maximal planar graph on n vertices could have in terms of n? Then in the last section we discuss the question by Hakimi, Schmeichel and Thomassen: what is the minimum number of Hamiltonian cycles that a maximal planar graph on n vertices could have, in terms of n? In Chapter 5, we look at general planar triangulations. Note that every maximal planar graph on n ≥ 3 vertices is a planar triangulation. In the first section we discuss general properties of planar triangulations and then end with Hamiltonian cycles in planar triangulations
Finding a Maximum Restricted -Matching via Boolean Edge-CSP
The problem of finding a maximum -matching without short cycles has
received significant attention due to its relevance to the Hamilton cycle
problem. This problem is generalized to finding a maximum -matching which
excludes specified complete -partite subgraphs, where is a fixed
positive integer. The polynomial solvability of this generalized problem
remains an open question. In this paper, we present polynomial-time algorithms
for the following two cases of this problem: in the first case the forbidden
complete -partite subgraphs are edge-disjoint; and in the second case the
maximum degree of the input graph is at most . Our result for the first
case extends the previous work of Nam (1994) showing the polynomial solvability
of the problem of finding a maximum -matching without cycles of length four,
where the cycles of length four are vertex-disjoint. The second result expands
upon the works of B\'{e}rczi and V\'{e}gh (2010) and Kobayashi and Yin (2012),
which focused on graphs with maximum degree at most . Our algorithms are
obtained from exploiting the discrete structure of restricted -matchings and
employing an algorithm for the Boolean edge-CSP.Comment: 20 pages, 2 figure
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