545 research outputs found
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 of certain generalized polygon trees
Let P(G) denote the chromatic polynomial of a graph G. Two graphs G and H are chromatically equivalent, written G ∼ H, if P(G) = P(H). Let g denote the family of all generalized polygon trees with three interior regions. Xu (1994) showed that g is a union of chromatic equivalence classes under the equivalence relation '∼'. In this paper, we determine infinitely many chromatic equivalence classes in g under '∼'. As a byproduct, we obtain a family of chromatically unique graphs established by Peng (1995)
Graph homomorphisms, the Tutte polynomial and “q-state Potts uniqueness”
We establish for which weighted graphs H homomorphism functions from multigraphs
G to H are specializations of the Tutte polynomial of G, answering a question
of Freedman, Lov´asz and Schrijver.
We introduce a new property of graphs called “q-state Potts uniqueness” and relate
it to chromatic and Tutte uniqueness, and also to “chromatic–flow uniqueness”,
recently studied by Duan, Wu and Yu.Ministerio de Educación y Ciencia MTM2005-08441-C02-0
Distinguishing graphs by their left and right homomorphism profiles
We introduce a new property of graphs called ‘q-state Potts unique-ness’ and relate it to chromatic and Tutte
uniqueness, and also to ‘chromatic–flow uniqueness’, recently studied by Duan, Wu and Yu.
We establish for which edge-weighted graphs H homomor-phism functions from multigraphs G to H are
specializations of the Tutte polynomial of G, in particular answering a question of Freed-man, Lovász and
Schrijver. We also determine for which edge-weighted graphs H homomorphism functions from
multigraphs G to H are specializations of the ‘edge elimination polynomial’ of Averbouch, Godlin and
Makowsky and the ‘induced subgraph poly-nomial’ of Tittmann, Averbouch and Makowsky.
Unifying the study of these and related problems is the notion of the left and right homomorphism profiles
of a graph.Ministerio de Educación y Ciencia MTM2008-05866-C03-01Junta de Andalucía FQM- 0164Junta de Andalucía P06-FQM-0164
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
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