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The degree sequence of a bipartite graph is the pair of lists each containing the degrees of the two parts and . For example, the complete bipartite graph K 3,5 has degree sequence (,,), (,,,,). Isomorphic bipartite graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a bipartite graph; in ...
Graph coloring enjoys many practical applications as well as theoretical challenges. Beside the classical types of problems, different limitations can also be set on the graph, or on the way a color is assigned, or even on the color itself. It has even reached popularity with the general public in the form of the popular number puzzle Sudoku ...
For a graph G, let χ(G) denote the chromatic number and Δ(G) the maximum degree of G.The list coloring number ch(G) satisfies the following properties.. ch(G) ≥ χ(G).A k-list-colorable graph must in particular have a list coloring when every vertex is assigned the same list of k colors, which corresponds to a usual k-coloring.
For instance, the drawing of the utility graph K 3,3 as the edges and long diagonals of a regular hexagon represents a 3-edge-coloring of the graph in this way. As Richter shows, a 3-regular simple bipartite graph, with a given Tait coloring, has a drawing of this type that represents the given coloring if and only if the graph is 3-edge-connected.
A list edge-coloring is a choice of a color for each edge, from its list of allowed colors; a coloring is proper if no two adjacent edges receive the same color. A graph G is k-edge-choosable if every instance of list edge-coloring that has G as its underlying graph and that provides at least k allowed colors for each edge of G has
An equitable coloring of the star K 1,5.. The star K 1,5 - a single central vertex connected to five others - is a complete bipartite graph, and therefore may be colored with two colors.
DSatur is known to be exact for bipartite graphs, [1] as well as for cycle and wheel graphs. [2] In an empirical comparison by Lewis in 2021, DSatur produced significantly better vertex colourings than the greedy algorithm on random graphs with edge probability p = 0.5 {\displaystyle p=0.5} , while in turn producing significantly worse ...
They proved () for bipartite graphs. [13] In case of regular bipartite graphs equality holds. Subcubic bipartite graphs admit an interval incidence coloring using four, five or six colors. They have also proved incidence 5-colorability can be decided in linear time for bipartite graphs with ∆(G) = 4.