On some colorings of a double graph
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Edge Coloring
Windmill graph
Brooks' theorem
Friendship graph
Critical graph
Critical graph
Edge Coloring
Counterexample
Windmill graph
Friendship graph
Petersen graph
Foster graph
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Edge Coloring
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In graph theory, an edge coloring of a graph is a coloring of the edges, meaning an assignment of colors to edges. Edge coloring can be described as function , where is the set of graph edges and is the set of natural numbers. Graph coloring has been studied as an algorithmic problem since the early 1970s. The main objective is to minimize the number of colors while coloring a graph. The smallest number of colors required to color a graph with specified conditions is called chromatic number of that graph and is denoted by . By a result of Holyer [1], the determination of the chromatic index is an hard optimization problem. The NP-hardness give rise to the necessity of using heuristic algorithms. In particular, we are interested in upper bounds for the chromatic index that can be efficiently realized by a coloring algorithm.
Edge Coloring
Graph Coloring
Fractional coloring
List coloring
Brooks' theorem
Complete coloring
Greedy coloring
Windmill graph
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The chromatic index X'(G) of a graph G is the minimun number of colors required to color the edges of G so that two adjacent edges receive different colors.In 1965,Vizing proved that if G is a graph of maximun degree △,then X'(G) is either △ or △ + 1.If G is a connected graph and X'(G-e) X'(G) exists in every edge(e) of G,it can be called a critical graph.In this paper,we prove several new results on chromatic index critical graphs.We also give a new upper bound on the size of chromatic index critical graphs of even order.
Edge Coloring
Critical graph
Connectivity
Friendship graph
Brooks' theorem
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The locating-chromatic number was introduced by Chartrand in 2002. The locating chromatic number of a graph is a combined concept between the coloring and partition dimension of a graph. The locating chromatic number of a graph is defined as the cardinality of the minimum color classes of the graph. In this paper, we discuss about the locating-chromatic number of shadow path graph and barbell graph containing shadow graph.
Windmill graph
Butterfly graph
Critical graph
Wheel graph
Friendship graph
Null graph
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Graph colorings are a major area of study in graph theory involving the constrained assignment of labels (colors) to vertices or edges. There are many types of colorings defined in the literature. The most common type of coloring is the proper vertex k-coloring which is defined as a vertex coloring from a set of k colors such that no two adjacent vertices share a common color. Our central focus in this paper is a variant of the proper vertex k-coloring problem, termed graceful coloring introduced by Gary Chartrand in 2015. A graceful k-coloring of an undirected connected graph G is a proper vertex coloring using k colors that induces a proper edge coloring, where the color for an edge (u,v) is the absolute value of the difference between the colors assigned to vertices u and v. In this work we find the graceful chromatic number, the minimum k for which a graph has a graceful k-coloring, for some well-known graphs and classes of graphs, such as diamond graph, Petersen graph, Moser spindle graph, Goldner-Harary graph, friendship graphs, fan graphs and others.
Fractional coloring
Edge Coloring
Graph Coloring
Complete coloring
Brooks' theorem
List coloring
Neighbourhood (mathematics)
Greedy coloring
Windmill graph
Friendship graph
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Edge Coloring
Multigraph
Critical graph
Friendship graph
Windmill graph
Fractional coloring
Brooks' theorem
Connectivity
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The locating-chromatic number of a graph was introduced by Chartrand et al. in 2002. The concept of the locating-chromatic number is a marriage between graph coloring and the notion of graph partition dimension. This concept is only for connected graphs. In [8], we extended this concept also for disconnected graphs. In this paper, we determine the locating- chromatic number of a graph with two components. In particular, we determine such values if the components are homogeneous and each component has locating-chromatic number 3.
Windmill graph
Friendship graph
Critical graph
Butterfly graph
Connected component
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In order to discuss the conjecture of total coloring of graph,the concept of total graph was proposed.In this paper,adjacent-distinguishing edge coloring of sveral particular graphs was studied,and the presisely chromatic numbers was obtained,a adjaceng-distinguishing edge coloring math was presented.
Edge Coloring
Brooks' theorem
Total coloring
Graph Coloring
Fractional coloring
List coloring
Complete coloring
Windmill graph
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Critical graph
Windmill graph
Friendship graph
Foster graph
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