Subgraph Of Minimum Degree Research Articles

A Note on Internal Partitions: The 5-Regular Case and Beyond

An internal or friendly partition of a graph is a partition of the vertex set into two nonempty sets so that every vertex has at least as many neighbours in its own class as in the other one. It has been shown that apart from finitely many counterexamples, every 3, 4 or 6-regular graph has an internal partition. In this note we focus on the 5-regular case and show that among the subgraphs of minimum degree at least 3 of 5-regular graphs, there are some which have small intersection. We also discuss the existence of internal partitions in some families of Cayley graphs, notably we determine all 5-regular Abelian Cayley graphs which do not have an internal partition.

A proof of a conjecture of Erdős, Faudree, Rousseau and Schelp on subgraphs of minimum degree k

Erdős, Faudree, Rousseau and Schelp observed the following fact for every fixed integer k≥2: Every graph on n≥k−1 vertices with at least (k−1)(n−k+2)+(k−22) edges contains a subgraph with minimum degree at least k. However, there are examples in which the whole graph is the only such subgraph. Erdős et al. conjectured that having just one more edge implies the existence of a subgraph on at most (1−εk)n vertices with minimum degree at least k, where εk>0 depends only on k. We prove this conjecture, using and extending ideas of Mousset, Noever and Škorić.

Smaller Subgraphs of Minimum Degree $k$

In 1990 Erdős, Faudree, Rousseau and Schelp proved that for $k \ge 2$, every graph with $n \ge k+1$ vertices and $(k-1)(n-k+2)+\binom{k-2}{2}+1$ edges contains a subgraph of minimum degree $k$ on at most $n-\sqrt{n/6k^3}$ vertices. They conjectured that it is possible to remove at least $\epsilon_k n$ many vertices and remain with a subgraph of minimum degree $k$, for some $\epsilon_k>0$. We make progress towards their conjecture by showing that one can remove at least order of $\Omega(n/\log n)$ many vertices.

How does the core sit inside the mantle?

The k-core, defined as the largest subgraph of minimum degree k, of the random graph G(n,p) has been studied extensively. In a landmark paper Pittel, Wormald and Spencer [Journal of Combinatorial Theory, Series B 67 (1996) 111–151] determined the threshold dk for the appearance of an extensive k-core. Here we derive a multi-type Galton-Watson branching process that describes precisely how the k-core is “embedded” into the random graph for any k≥3 and any fixed average degree d=np>dk. This generalises prior results on, e.g., the internal structure of the k-core.

Trees in greedy colorings of hypergraphs

It is a well-known proposition that every graph of chromatic number larger than t contains every tree with t edges. The ‘standard’ reasoning is that such a graph must contain a subgraph of minimum degree at least t . Bohman, Frieze, and Mubayi noticed that, although this argument does not work for hypergraphs, it is still possible that the proposition holds for hypergraphs as well. Indeed, Loh recently proved that every uniform hypergraph of chromatic number larger than t contains every hypertree with t edges. Here we observe that the basic property of the well-known greedy algorithm immediately implies a much more general result (with a conceptually simpler proof): if the greedy algorithm colors the vertices of an r -uniform hypergraph with more than t colors then the hypergraph contains every r -uniform hypertree with t edges.

A general critical condition for the emergence of a giant component in random graphs with given degrees

In this contribution, we investigate the giant component problem in random graphs with a given degree sequence. We generalize the critical condition of Molloy and Reed [Molloy, M., and B. Reed, A critical point for random graphs with given degree sequence, Random Structures Algorithms 6 (1995), 161-179], which determines the existence of a giant component in such a random graph, in order to include degree sequences with heavy tails. We show that the quantity which determines the existence of a giant component is the value of the smallest fixed point inside the interval [0, 1] of the generating function F(s)=∑i⩾1δisi−1, where δi is the asymptotic proportion of the total degree contained in vertices of degree i. Moreover, we show that this quantity also determines the existence of a core (i.e., the maximal subgraph of minimum degree at least 2) that has linear total degree.

Generalized trees related with tree metrics

Abstract Graphs without induced subgraphs of minimum degree at least 3 have been considered in matroid theory (see, e.g., Welsh 1976, ch. 14 or Aigner 1979, ch. 7) and in some domains of applications (Todd 1989, Guenoche and Leclerc 1999). Maximal such graphs were characterized in Todd's paper. They are called here 2d-trees. A subclass of 2d-trees consists of the classical 2-trees (Pippert and Beineke 1969), whose interesting status among chordal, series parallel and 2-connected graphs has been extensively studied in the literature (Arnborg and proskurowski 1989, Borie, Parker and Tovey 1991, Rose 1974, and others). We consider here valued 2d-trees and 2-trees with a given finite vertex set X.

Decomposing graphs under degree constraints

We prove a conjecture of C. Thomassen: If s and t are non-negative integers, and if G is a graph with minimum degree s + t + 1, then the vertex set of G can be partitioned into two sets which induce subgraphs of minimum degree at least s and t, respectively. © 1996 John Wiley & Sons, Inc.