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In graph theory, a k-degenerate graph is an undirected graph in which every subgraph has at least one vertex of degree at most k: that is, some vertex in the subgraph touches k or fewer of the subgraph's edges. The degeneracy of a graph is the smallest value of k for which it is k-degenerate.
Any complete graph is a core. A cycle of odd length is a core. A graph is a core if and only if the core of is equal to . Every two cycles of even length, and more generally every two bipartite graphs are hom-equivalent. The core of each of these graphs is the two-vertex complete graph K 2.
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.
The remaining blue vertices form the 2-core of the graph. Least fixed point based logics of graphs extend the first-order logic of graphs by allowing predicates (properties of vertices or tuples of vertices) defined by special fixed-point operators. This kind of definition begins with an implication, a formula stating that when certain values ...
In graph theory, a part of mathematics, a k-partite graph is a graph whose vertices are (or can be) partitioned into k different independent sets. Equivalently, it is a graph that can be colored with k colors, so that no two endpoints of an edge have the same color. When k = 2 these are the bipartite graphs, and when k = 3 they are called the ...
For k = 3, every k-critical graph (that is, every odd cycle) can be generated as a k-constructible graph such that all of the graphs formed in its construction are also k-critical. For k = 8 , this is not true: a graph found by Catlin (1979) as a counterexample to Hajós's conjecture that k -chromatic graphs contain a subdivision of K k , also ...
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A k-factor of a graph is a spanning k-regular subgraph, and a k-factorization partitions the edges of the graph into disjoint k-factors. A graph G is said to be k-factorable if it admits a k-factorization. In particular, a 1-factor is a perfect matching, and a 1-factorization of a k-regular graph is a proper edge coloring with k colors.