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Showing 1 to 20 of 43 for “"chromatic number"”.
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Two graph classes with bounded chromatic number
A class of graphs is said to be $\chi$-bounded with binding function $f$ if for every such graph $G$, it satisfies $\chi(G) \leq f(\omega(G)$, and polynomially $\chi$-bounded if $f$ is a polynomial. It was conjectured that chair-free graphs are perfectly divisible, and hence admit a quadratic …
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The difficulty of approximating the chromatic number for random composite graphs
<p>"Combinatorial Optimization is an important class of techniques for solving Combinatorial Problems. Many practical problems are Combinatorial Problems, such as the Traveling Salesman Problem (TSP) and Composite Graph Coloring Problem (CGCP). Unfortunately, both of these problems are ��P-complete …
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Topological Approaches to Chromatic Number and Box Complex Analysis of Partition Graphs
Determining the chromatic number of the partition graph P(33) poses a considerable challenge. We can bound it to 4 ≤ χ(P(33)) ≤ 6, with exhaustive search confirming χ(P(33)) = 6. A potential mathematical proof strategy for this equality involves identifying a Z2-invariant S4 with non-trivial …
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On the Attainability of Upper Bounds for the Circular Chromatic Number of <em>K</em><sub>4</sub>-Minor-Free Graphs.
… an edge of <em>G</em>. We say that the circular chromatic number of <em>G</em>, denoted <em>χ<sub>c</sub></em>(<em>G</em>), is equal to the smallest <em>k</em>/<em>d</em> where a <em>k</em>/<em>d</em> -coloring exists. In [6], Pan and Zhu have given a function <em>μ</em>(<em>g</em>) that gives an …
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Dynamic coloring of graphs
… subjects of dynamic colorings, we compare the chromatic number and dynamic chromatic number, and we study some problems unique to dynamic colorings. Also, we introduce and briefly study a generalization of dynamic coloring.;The interesting subjects of colorings we consider are the chromatic …
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The Chromatic Structure of Dense Graphs
… macroscopic structure. We primarily consider the chromatic structure: whether a graph has or is close to having some (low) chromatic number. Chapter 2 is the slight exception. We consider an induced version of the classical Turán problem. Introduced by Loh, Tait, Timmons, and Zhou, the induced …
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Chromatic Thresholds of Regular Graphs with Small Cliques
<p>The chromatic threshold of a class of graphs is the value <em>θ</em> such that any graph in this class with a minimum degree greater than <em>θn</em> has a bounded chromatic number. Several important results related to the chromatic threshold of triangle-free graphs have been reached in the last …
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Results in Ramsey theory and extremal graph theory
… work with Vojtěch Dvořák, we consider the Ramsey number $R(F_n)$ of the fan graph $F_n$, a graph consisting of $n$ triangles which all share a common vertex. Chen, Yu and Zhao showed that $\frac{9}{2}n-5 \leq R(F_n) \leq \frac{11}{2}n+6$. We build on the techniques that they used to prove the …
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Jogos combinatórios em grafos: jogo Timber e jogo de Coloração
… when played in trees. We determine the number of P-positions in three caterpillar families and a lower bound for the number of P-positions in any caterpillar. Moreover, we prove that a tree has P-positions if, and only if, it has an even number of edges. In the coloring game, Alice and …
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Extremal problems involving forbidden subgraphs
… of graph coloring. We determine harmonious chromatic number of trees with large maximum degree and show upper bounds of $r$-dynamic chromatic number of graphs in terms of other parameters. In Chapter 4, we consider how dense a hypergraph can be when we forbid some subgraphs. In particular, …
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Edge coloring of simple graphs and edge -face coloring of simple plane graphs
… Denote chie( G), chief(G), Delta( G) the edge chromatic number, the edge-face chromatic number and the maximum degree of G, respectively. We prove that chi ef(G) = chie( G) = Delta(G) for any 2-connected simple plane graph G with Delta(G) ≥ 24.
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Fractional Chromatic Numbers and Spectra of Graphs
… mainly comes from my recent study of fractional chromatic numbers of graphs, spectra of edge-independent random graphs, Laplacian spectra of hypergraphs, and loose Laplacian spectra of random hypergraphs.</p> <p>For a graph $G$, let $\chi_f(G)$ be the fractional chromatic number of $G$. Based on …
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Extremal problems in graph theory
… of small radius and find the domination number of the Kneser graph $K(n,k)$ when $n\ge{3\over4}k\sp2\pm k,$ depending on whether k is even or odd. The path chromatic number $\chi\sb{P}(G)$ of a graph G is the least number of colors with which the vertices of G can be colored so that each …
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Problems in Graph Coloring and Graph Structure
… be the intersection graph of F . We studied the chromatic number of the complement of G. We also studied the transversal number of F , where the transversal number is the minimum size of a set of points that intersects all convex sets in F .
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Small cycle cover, group coloring with related problems
… group coloring in 1992 and proved that the group chromatic number for every planar graph is at most 6. It is shown that the bound 6 can be decreased to 5. Jaeger, Linial, Payan and Tarsi also proved that the group chromatic number for every planar graph with girth at least 4 is at most 4. Chapters …
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Matchings, Connectivity, and Eigenvalues in Regular Graphs
… we obtain the best lower bound for the matching number over $n$-vertex connected regular graphs in terms of edge-connectedness and determine when the matching number is minimized. We also establish the best upper bound for the number of cut-edges over $n$-vertex connected odd regular graphs and …
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D-colorable digraphs with large girth
… arbitrarily large girth and arbitrarily large chromatic number. This result, along with its proof, has had a number of descendants (D. Bokal, G. Fijavz, M. Juvan, P.M. Kayll and B. Mohar, <italic>The circular chromatic number of a digraph</italic>, J. Graph Theory <bold>46</bold> (2004), …
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D-colorable digraphs with large girth
… arbitrarily large girth and arbitrarily large chromatic number. This result, along with its proof, has had a number of descendants (D. Bokal, G. Fijavz, M. Juvan, P.M. Kayll and B. Mohar, <italic>The circular chromatic number of a digraph</italic>, J. Graph Theory <bold>46</bold> (2004), …
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Coloring and constructing (hyper)graphs with restrictions
… Gyárfás, and Schelp concerning the minimum number of edges in a “nearly bipartite” 4-critical graph. In Chapter 3 we consider coloring and list-coloring graphs and hypergraphs with few edges and no small cycles. We prove two main results. If a bipartite graph has maximum average degree at …
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Restrained and Other Domination Parameters in Complementary Prisms.
… the known results addressing the domination number and the total domination number of complementary prisms. After this, we will present our main results, namely, results on the restrained domination number of complementary prisms. Subsequently results on the <em>distance</em> - <em>k</em> …
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