67 research outputs found
Chromaticity of Certain Bipartite Graphs
Since the introduction of the concepts of chromatically unique graphs and chromatically
equivalent graphs, numerous families of such graphs have been obtained.
The purpose of this thesis is to continue with the search of families of
chromatically unique bipartite graphs.
In Chapters 1 and 2, we define the concept of graph colouring, the associated
chromatic polynomial and some properties of a chromatic polynomial. We also
give some necessary conditions for graphs that are chromatically unique or chromatically
equivalent. We end this chapter by stating some known results on the
chromaticity of bipartite graphs, denoted as K(p,q)
Chromaticity of Certain 2-Connected Graphs
Since the introduction of the concepts of chromatically unique graphs and chromatically
equivalent graphs, many families of such graphs have been obtained.
In this thesis, we continue with the search of families of chromatically unique
graphs and chromatically equivalent graphs.
In Chapter 1, we define the concept of graph colouring, the associated chromatic
polynomial and some properties of a chromatic polynomial. We also give some
necessary conditions for graphs that are chromatically unique or chromatically
equivalent.
Chapter 2 deals with the chromatic classes of certain existing 2-connected (n, n + 1,)-graphs for z = 0, 1, 2 and 3. Many families of chromatically unique graphs and
chromatically equivalent graphs of these classes have been obtained. At the end
of the chapter, we re-determine the chromaticity of two families of 2-connected
(n, n + 3)-graphs with at least two triangles. Our main results in this thesis are presented in Chapters 3, 4 and 5. In Chapter
3, we classify all the 2-connected (n, n + 4)-graphs wit h at least four triangles . In
Chapter 4 , we classify all the 2-connected (n, n + 4)-graphs wit h t hree triangles
and one induced 4-cycle. In Chapter 5, we classify all the 2-connected (n, n + 4)graphs
with three triangles and at least two induced 4-cycles . In each chapter, we
obtain new families of chromatically unique graphs and chromatically equivalent
graphs.
We end the thesis by classifying all the 2-connected (n, n + 4)-graphs with exactly
three triangles. We also determine the chromatic polynomial of all these graphs.
The determination of the chromaticity of most classes of these graphs is left as
an open problem for future research
Chromatic Equivalence Classes and Chromatic Defining Numbers of Certain Graphs
There are two parts in this dissertation: the chromatic equivalence classes and
the chromatic defining numbers of graphs.
In the first part the chromaticity of the family of generalized polygon trees with
intercourse number two, denoted by Cr (a, b; c, d), is studied. It is known that
Cr( a, b; c, d) is a chromatic equivalence class if min {a, b, c, d} ≥ r+3. We consider
Cr( a, b; c, d) when min{ a, b, c, d} ≤ r + 2. The necessary and sufficient conditions
for Cr(a, b; c, d) with min {a, b, c, d} ≤ r + 2 to be a chromatic equivalence class
are given. Thus, the chromaticity of Cr (a, b; c, d) is completely characterized.
In the second part the defining numbers of regular graphs are studied. Let
d(n, r, X = k) be the smallest value of defining numbers of all r-regular graphs
of order n and the chromatic number equals to k. It is proved that for a given
integer k and each r ≥ 2(k - 1) and n ≥ 2k, d(n, r, X = k) = k - 1. Next,
a new lower bound for the defining numbers of r-regular k-chromatic graphs
with k < r < 2( k - 1) is found. Finally, the value of d( n , r, X = k) when
k < r < 2(k - 1) for certain values of n and r is determined
The chromaticity of wheels with a missing spoke II
AbstractIn the previous paper, it was shown that the graph Un + 1 obtained from the wheel Wn + 1 by deleting a spoke is uniquely determined by its chromatic polynomial if n ⩾ 3 is odd. In this paper, we show that the result is also true for even n ⩾ 4 except when n = 6 in which case, the graph W given in the paper is the only graph having the same chromatic polynomial as that of U7. The relevant tool is the notion of nearly uniquely colorable graph
Exact Results on Potts Model Partition Functions in a Generalized External Field and Weighted-Set Graph Colorings
We present exact results on the partition function of the -state Potts
model on various families of graphs in a generalized external magnetic
field that favors or disfavors spin values in a subset of
the total set of possible spin values, , where and are
temperature- and field-dependent Boltzmann variables. We remark on differences
in thermodynamic behavior between our model with a generalized external
magnetic field and the Potts model with a conventional magnetic field that
favors or disfavors a single spin value. Exact results are also given for the
interesting special case of the zero-temperature Potts antiferromagnet,
corresponding to a set-weighted chromatic polynomial that counts
the number of colorings of the vertices of subject to the condition that
colors of adjacent vertices are different, with a weighting that favors or
disfavors colors in the interval . We derive powerful new upper and lower
bounds on for the ferromagnetic case in terms of zero-field
Potts partition functions with certain transformed arguments. We also prove
general inequalities for on different families of tree graphs.
As part of our analysis, we elucidate how the field-dependent Potts partition
function and weighted-set chromatic polynomial distinguish, respectively,
between Tutte-equivalent and chromatically equivalent pairs of graphs.Comment: 39 pages, 1 figur
Chromaticity Of Certain K4-Homeomorphs
The chromaticity of graphs is the term used referring to the question of chromatic
equivalence and chromatic uniqueness of graphs. Since the arousal of the
interest on the chromatically equivalent and chromatically unique graphs, various
concepts and results under the said areas of research have been discovered and
many families of such graphs have been obtained. The purpose of this thesis is to
contribute new results on the chromatic equivalence and chromatic uniqueness
of graphs, specifically, K4-homeomorphs
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