|
Interval Edge Coloring
In graph theory, interval edge coloring is a type of edge coloring in which edges are labeled by the integers in some interval, every integer in the interval is used by at least one edge, and at each vertex the labels that appear on incident edges form a consecutive set of distinct numbers. History The concept of consecutive edge-coloring was introduced with the terminology 'interval edge coloring' by Asratian and Kamalian in 1987 in their paper "Interval colorings of edges of a multigraph". Since interval edge coloring of graphs was introduced mathematicians have been investigating the existence of interval edge colorable graphs as not all graphs allow interval edge coloring. A simple family of graphs that allows interval edge coloring is complete graph of even order and a counter example of family of graphs includes complete graphs of odd order. The smallest graph that doesnot allow interval colorability.There are even graphs discovered with 28 vertices and maximum degree 21 that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L(h, K)-coloring
In graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ..., a L(''h'', ''k'')-labelling, L(''h'', ''k'')-coloring or sometimes L(''p'', ''q'')-coloring is a (proper) vertex coloring in which every pair of adjacent vertices has color numbers that differ by at least ''h'', and any nodes connected by a 2 length path have their colors differ by at least ''k''. The parameters, ''h'' and ''k'' are understood to be non-negative integers. The problem originated from a channel assignment problem in radio networks. The span of an L(''h'', ''k'')-labelling, ρh,k(G) is the difference between the largest and the smallest assigned frequency. The goal of the L(''h'', ''k'')-labelling problem is usually to find a labelling with minimum span. For a given graph, the minimum span over all poss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sum Coloring
In graph theory, a sum coloring of a graph is a labeling of its vertices by positive integers, with no two adjacent vertices having equal labels, that minimizes the sum of the labels. The minimum sum that can be achieved is called the chromatic sum of the graph. Chromatic sums and sum coloring were introduced by Supowit in 1987 using non-graph-theoretic terminology, and first studied in graph theoretic terms by Ewa Kubicka (independently of Supowit) in her 1989 doctoral thesis. Obtaining the chromatic sum may require using more distinct labels than the chromatic number of the graph, and even when the chromatic number of a graph is bounded, the number of distinct labels needed to obtain the optimal chromatic sum may be arbitrarily large. Computing the chromatic sum is NP-hard. However it may be computed in linear time for trees and pseudotrees, and in polynomial time for outerplanar graphs. There is a constant-factor approximation algorithm for interval graphs and for biparti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonious Coloring
In graph theory, a harmonious coloring is a (proper) vertex coloring in which every pair of colors appears on ''at most'' one pair of adjacent vertices. It is the opposite of the complete coloring, which instead requires every color pairing to occur ''at least'' once. The harmonious chromatic number of a graph is the minimum number of colors needed for any harmonious coloring of . Every graph has a harmonious coloring, since it suffices to assign every vertex a distinct color; thus . There trivially exist graphs with (where is the chromatic number); one example is any path of , which can be 2-colored but has no harmonious coloring with 2 colors. Some properties of : :\chi_(T_) = \left\lceil\frac\right\rceil, where is the complete -ary tree with 3 levels. (Mitchem 1989) Harmonious coloring was first proposed by Harary and Plantholt (1982). Still very little is known about it. See also * Complete coloring In graph theory, a complete coloring is a vertex coloring in whi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Coloring
In the mathematical field of graph theory, a star coloring of a graph is a (proper) vertex coloring in which every path on four vertices uses at least three distinct colors. Equivalently, in a star coloring, the induced subgraphs formed by the vertices of any two colors has connected components that are star graphs. Star coloring has been introduced by . The star chromatic number of is the fewest colors needed to star color . One generalization of star coloring is the closely related concept of acyclic coloring, where it is required that every cycle uses at least three colors, so the two-color induced subgraphs are forests. If we denote the acyclic chromatic number of a graph by , we have that , and in fact every star coloring of is an acyclic coloring. The star chromatic number has been proved to be bounded on every proper minor closed class by . This results was further generalized by to all low-tree-depth colorings (standard coloring and star coloring being low-tree- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acyclic Coloring
In graph theory, an acyclic coloring is a (proper) Graph coloring, vertex coloring in which every 2-chromatic subgraph is Glossary of graph theory, acyclic. The acyclic chromatic number of a Graph (discrete mathematics), graph is the fewest colors needed in any acyclic coloring of . Acyclic coloring is often associated with graphs Graph embedding, embedded on non-plane surfaces. Upper bounds A(''G'') ≤ 2 if and only if ''G'' is acyclic. Bounds on A(''G'') in terms of Δ(''G''), the Glossary of graph theory, maximum degree of ''G'', include the following: * A(''G'') ≤ 4 if Δ(''G'') = 3. * A(''G'') ≤ 5 if Δ(''G'') = 4. * A(''G'') ≤ 7 if Δ(''G'') = 5. * A(''G'') ≤ 12 if Δ(''G'') = 6. A milestone in the study of acyclic coloring is the following affirmative answer to a conjecture of Grünbaum: :Theorem A(''G'') ≤ 5 if ''G'' is planar graph. introduced acyclic coloring and acyclic chromatic number, and conjectured the result in the above theorem. Borodin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radio Coloring
In graph theory, a branch of mathematics, a radio coloring of an undirected graph is a form of graph coloring in which one assigns positive integer labels to the graphs such that the labels of adjacent vertices differ by at least two, and the labels of vertices at distance two from each other differ by at least one. Radio coloring was first studied by , under a different name, -labeling. It was called radio coloring by Frank Harary because it models the problem of channel assignment in radio broadcasting, while avoiding electromagnetic interference between radio stations that are near each other both in the graph and in their assigned channel frequencies. The span of a radio coloring is its maximum label, and the radio coloring number of a graph is the smallest possible span of a radio coloring.. See in particular Section 3, "Radio coloring". For instance, the graph consisting of two vertices with a single edge has radio coloring number 3: it has a radio coloring with one vertex la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Coloring
In graph theory, total coloring is a type of graph coloring on the vertices and edges of a graph. When used without any qualification, a total coloring is always assumed to be ''proper'' in the sense that no adjacent edges, no adjacent vertices and no edge and either endvertex are assigned the same color. The total chromatic number χ″(''G'') of a graph ''G'' is the fewest colors needed in any total coloring of ''G''. The total graph ''T'' = ''T''(''G'') of a graph ''G'' is a graph such that (i) the vertex set of ''T'' corresponds to the vertices and edges of ''G'' and (ii) two vertices are adjacent in ''T'' if and only if their corresponding elements are either adjacent or incident in ''G''. Then total coloring of ''G'' becomes a (proper) vertex coloring of ''T''(''G''). A total coloring is a partitioning of the vertices and edges of the graph into total independent sets. Some inequalities for χ″(''G''): # χ″(''G'') ≥ Δ(''G'') + 1. # χ″(''G'') ≤ Δ(''G'') + 1026 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incidence Coloring
In graph theory, the act of coloring generally implies the assignment of labels to vertices, edges or faces in a graph. The incidence coloring is a special graph labeling where each incidence of an edge with a vertex is assigned a color under certain constraints. Definitions Below ''G'' denotes a simple graph with non-empty vertex set (non-empty) ''V''(''G''), edge set ''E''(''G'') and maximum degree Δ(''G''). Definition. An incidence is defined as a pair (''v'', ''e'') where v\in V(G) is an end point of e\in E(G). In simple words, one says that vertex ''v'' is incident to edge ''e''. Two incidences (''v'', ''e'') and (''u'', ''f'') are said to be adjacent or neighboring if one of the following holds: * ''v'' = ''u'', ''e'' ≠ ''f'' * ''e'' = ''f'', ''v'' ≠ ''u'' * ''e'' = , ''f'' = and ''v'' ≠ ''w''. Definition. Let ''I''(''G'') be the set of all incidences of ''G''. An incidence coloring of ''G'' is a function c: I(G)\to\N that takes distinct values on adjacent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Defective Coloring
In graph theory, a mathematical discipline, coloring refers to an assignment of colours or labels to vertices, edges and faces of a graph. Defective coloring is a variant of proper vertex coloring. In a proper vertex coloring, the vertices are coloured such that no adjacent vertices have the same colour. In defective coloring, on the other hand, vertices are allowed to have neighbours of the same colour to a certain extent. (See here for Glossary of graph theory) History Defective coloring was introduced nearly simultaneously by Burr and Jacobson, Harary and Jones and Cowen, Cowen and Woodall. Surveys of this and related colorings are given by Marietjie Frick. Cowen, Cowen and Woodall focused on graphs embedded on surfaces and gave a complete characterization of all ''k'' and ''d'' such that every planar is (''k'', ''d'')-colorable. Namely, there does not exist a ''d'' such that every planar graph is (1, ''d'')- or (2, ''d'')-colorable; there exist planar graphs which are not (3, 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge Coloring
In graph theory, an edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring. The edge-coloring problem asks whether it is possible to color the edges of a given graph using at most different colors, for a given value of , or with the fewest possible colors. The minimum required number of colors for the edges of a given graph is called the chromatic index of the graph. For example, the edges of the graph in the illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three. By Vizing's theorem, the number of colors needed to edge color a simple graph is either its maximum degree or . For some graphs, such as bipartite graphs and high-degree planar graphs, the number of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path Coloring
In graph theory, path coloring usually refers to one of two problems: * The problem of coloring a (multi)set of paths R in graph G, in such a way that any two paths of R which share an edge in G receive different colors. Set R and graph G are provided at input. This formulation is equivalent to vertex coloring the ''conflict graph'' of set R, i.e. a graph with vertex set R and edges connecting all pairs of paths of R which are not edge-disjoint with respect to G. * The problem of coloring (in accordance with the above definition) any chosen (multi)set R of paths in G, such that the set of pairs of end-vertices of paths from R is equal to some set or multiset I, called a ''set of requests''. Set I and graph G are provided at input. This problem is a special case of a more general class of graph routing problems, known as call scheduling. In both the above problems, the goal is usually to minimise the number of colors used in the coloring. In different variants of path coloring, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
