20 research outputs found

    Chromaticity of Certain Bipartite Graphs

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    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 Bipartite Graphs With Three And Four Edges Deleted

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    Graphs are a set of vertices and edges. All vertices may or may not be joint. Vertex coloring is the coloring of a graph with a fixed number of colors such that adjacent vertices are of different colors. Graf adalah satu set titik dengan satu set garisan yang menghubungkan titik-titik. Semua titik mungkin dihubungkan dengan garisan

    Graphs determined by polynomial invariants

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    AbstractMany polynomials have been defined associated to graphs, like the characteristic, matchings, chromatic and Tutte polynomials. Besides their intrinsic interest, they encode useful combinatorial information about the given graph. It is natural then to ask to what extent any of these polynomials determines a graph and, in particular, whether one can find graphs that can be uniquely determined by a given polynomial. In this paper we survey known results in this area and, at the same time, we present some new results

    3-Maps And Their Generalizations

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    A 3-map is a 3-region colorable map. They have been studied by Craft and White in their paper 3-maps. This thesis introduces topological graph theory and then investigates 3-maps in detail, including examples, special types of 3-maps, the use of 3-maps to find the genus of special graphs, and a generalization known as n-maps

    A Little Statistical Mechanics for the Graph Theorist

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    In this survey, we give a friendly introduction from a graph theory perspective to the q-state Potts model, an important statistical mechanics tool for analyzing complex systems in which nearest neighbor interactions determine the aggregate behavior of the system. We present the surprising equivalence of the Potts model partition function and one of the most renowned graph invariants, the Tutte polynomial, a relationship that has resulted in a remarkable synergy between the two fields of study. We highlight some of these interconnections, such as computational complexity results that have alternated between the two fields. The Potts model captures the effect of temperature on the system and plays an important role in the study of thermodynamic phase transitions. We discuss the equivalence of the chromatic polynomial and the zero-temperature antiferromagnetic partition function, and how this has led to the study of the complex zeros of these functions. We also briefly describe Monte Carlo simulations commonly used for Potts model analysis of complex systems. The Potts model has applications as widely varied as magnetism, tumor migration, foam behaviors, and social demographics, and we provide a sampling of these that also demonstrates some variations of the Potts model. We conclude with some current areas of investigation that emphasize graph theoretic approaches. This paper is an elementary general audience survey, intended to popularize the area and provide an accessible first point of entry for further exploration.Comment: 30 pages, 3 figure

    Dense graphs have K3,t minors

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    AbstractLet K3,t∗ denote the graph obtained from K3,t by adding all edges between the three vertices of degree t in it. We prove that for each t≥6300 and n≥t+3, each n-vertex graph G with e(G)>12(t+3)(n−2)+1 has a K3,t∗-minor. The bound is sharp in the sense that for every t, there are infinitely many graphs G with e(G)=12(t+3)(|V(G)|−2)+1 that have no K3,t-minor. The result confirms a partial case of the conjecture by Woodall and Seymour that every (s+t)-chromatic graph has a Ks,t-minor

    Subject Index Volumes 1–200

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