Color-Critical Graphs Have Logarithmic Circumference
نویسندگان
چکیده
A graph G is k-critical if every proper subgraph of G is (k − 1)-colorable, but the graph G itself is not. We prove that every k-critical graph on n vertices has a cycle of length at least logn/(100 logk), improving a bound of Alon, Krivelevich and Seymour from 2000. Examples of Gallai from 1963 show that the bound cannot be improved to exceed 2(k − 1) logn/ log(k − 2). We thus settle the problem of bounding the minimal circumference of k-critical graphs, raised by Dirac in 1952 and Kelly and Kelly in 1954.
منابع مشابه
Topology-Hiding Computation Beyond Logarithmic Diameter
A distributed computation in which nodes are connected by a partial communication graph is called topology-hiding if it does not reveal information about the graph (beyond what is revealed by the output of the function). Previous results [Moran, Orlov, Richelson; TCC’15] have shown that topology-hiding computation protocols exist for graphs of logarithmic diameter (in the number of nodes), but ...
متن کاملA growth chart for premature and other infants.
Gairdner, D., and Pearson, J. (1971). Archives of Disease in Childhood, 46, 783. A growth chart for premature and other infants. The growth chart covers the period from 28 weeks of fetal life to age 2 years, and allows weight, length, and head circumference to be charted. The time scale is logarithmic, providing the convenience of an extended scale for the earlier months of life, and giving the...
متن کاملColor-Critical Graphs and Hypergraphs with Few Edges: A Survey
A hypergraph is color-critical if deleting any edge or vertex reduces the chromatic number; a color-critical hypergraph with chromatic number k is k-critical. Every k-chromatic hypergraph contains a k-critical hypergraph, so one can study chromatic number by studying the structure of k-critical (hyper)graphs. There is vast literature on k-critical graphs and hypergraphs. Many references can be ...
متن کاملGraph Isomorphism for Colored Graphs with Color Multiplicity Bounded by 3
The colored graph isomorphism problem is a restricted version of the general graph isomorphism (GI) problem that involves deciding the existence of a color preserving isomorphism between a pair of colored graphs. In this report, we study this problem for graphs whose color multiplicity is bounded by 3 (3-GI). We begin by formally defining the colored graph isomorphism problem and setup our nota...
متن کاملThe locating-chromatic number for Halin graphs
Let G be a connected graph. Let f be a proper k -coloring of G and Π = (R_1, R_2, . . . , R_k) bean ordered partition of V (G) into color classes. For any vertex v of G, define the color code c_Π(v) of v with respect to Π to be a k -tuple (d(v, R_1), d(v, R_2), . . . , d(v, R_k)), where d(v, R_i) is the min{d(v, x)|x ∈ R_i}. If distinct vertices have distinct color codes, then we call f a locat...
متن کامل