Large Induced Subgraphs with $k$ Vertices of Almost Maximum Degree

Large induced subgraphs with equated maximum degree

For a graph G, denote by fk(G) the smallest number of vertices that must be deleted from G so that the remaining induced subgraph has its maximum degree shared by at least k vertices. It is not difficult to prove that there are graphs for which already f2(G) ≥ √ n(1− o(1)), where n is the number of vertices of G. It is conjectured that fk(G) = Θ( √ n) for every fixed k. We prove this for k = 2,...

Sizes of graphs with induced subgraphs of large maximum degree

Graphs with n + k vertices in which every set of n +j vertices induce a subgraph of maximum degree at least n are considered. For j = 1 and for k fairly small compared to n, we determine the minimum number of edges in such graphs. In investigating the size Ramsey number of a star KI,, versus a triangle K3, Erdiis [ 1,2] conjectured that for n 3 3 any graph with no more than (2”:‘) (‘;) 1 edges ...

Disjoint subgraphs of large maximum degree

Maximum k -regular induced subgraphs

Independent sets, induced matchings and cliques are examples of regular induced subgraphs in a graph. In this paper, we prove that finding a maximum cardinality k-regular induced subgraph is an NP-hard problem for any value of k. We propose a convex quadratic upper bound on the size of a k-regular induced subgraph and characterize those graphs for which this bound is attained. Finally, we exten...

k-forested choosability of graphs with bounded maximum average degree

A proper vertex coloring of a simple graph is $k$-forested if the graph induced by the vertices of any two color classes is a forest with maximum degree less than $k$. A graph is $k$-forested $q$-choosable if for a given list of $q$ colors associated with each vertex $v$, there exists a $k$-forested coloring of $G$ such that each vertex receives a color from its own list. In this paper, we prov...