Minimum degree conditions for large subgraphs

Much of extremal graph theory has concentrated either on finding very small subgraphs of a large graph (such as Turán's theorem [Turán, P., On an extremal problem in graph theory (in Hungarian), Matematiko Fizicki Lapok 48 (1941), 436–452]) or on finding spanning subgraphs (such as Dirac's theorem [Dirac, G.A., Some theorems on abstract graphs, Proc. London Math. Soc. s3-2 (1952), 69–81] or more recently work of Komlós, Sárközy and Szemerédi [Komlós, J., G. N. Sárközy and E. Szemerédi, On the square of a Hamiltonian cycle in dense graphs, Random Struct. Algorithms 9 (1996), 193-211; Komlós, J., G. N. Sárközy and E. Szemerédi, Proof of the Seymour Conjecture for large graphs, Ann. Comb. 2 (1998), 43–60] towards a proof of the Pósa-Seymour conjecture). Only a few results give conditions to obtain some intermediate-sized subgraph. We contend that this neglect is unjustified. To support our contention we focus on the illustrative case of minimum degree conditions which guarantee squared-cycles of various lengths, but also offer results, conjectures and comments on other powers of paths and cycles, generalisations thereof, and hypergraph variants.     

On groups with large character degree sums

Let G be a finite group, and let T(G) be the sum of all complex irreducible character degrees of G. Define ${f(G) = \frac{T(G)}{|G|}}$ . In this paper, we show that if G is a finite group and ${f(G) > \frac{4}{15}}$ , then G is solvable.     

Connected subgraphs with small degree sums in 3‐connected planar graphs

It is well‐known that every planar graph has a vertex of degree at most five. Kotzig proved that every 3‐connected planar graph has an edge xy such that deg(x) + deg (y) ≤ 13. In this article, considering a similar problem for the case of three or more vertices that induce a connected subgraph, we show that, for a given positive integer t, every 3‐connected planar graph G with |V(G)| ≥ t has a connected subgraph H of order t such that Σ_x∈V(H) deg_G(x) ≤ 8t − 1. As a tool for proving this result, we consider decompositions of 3‐connected planar graphs into connected subgraphs of order at least t and at most 2t − 1. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 191–203, 1999     

Long paths and cycles in random subgraphs of graphs with large minimum degree

For a given finite graph G of minimum degree at least k, let G_p be a random subgraph of G obtained by taking each edge independently with probability p. We prove that (i) if for a function that tends to infinity as k does, then G_p asymptotically almost surely contains a cycle (and thus a path) of length at least , and (ii) if , then G_p asymptotically almost surely contains a path of length at least k. Our theorems extend classical results on paths and cycles in the binomial random graph, obtained by taking G to be the complete graph on k + 1 vertices. © Wiley Periodicals, Inc. Random Struct. Alg., 46, 320–345, 2015     

Long cycles in graphs with large degree sums and neighborhood unions

We present and prove several results concerning the length of longest cycles in 2-connected or 1-tough graphs with large degree sums. These results improve many known results on long cycles in these graphs. We also consider the sharpness of the results and discuss some possible strengthenings. © 1996 John Wiley & Sons, Inc.     

Complete subgraphs are elusive

We would like to decide whether a graph with a given vertex set has a certain property. We do this by probing the pairs of vertices one by one, i.e., asking whether a given pair of vertices is an edge or not. At each stage we make full use of the information we have up to that point. If there is an algorithm (a sequence of probes depending on the previous information) that allows us to come to a decision before checking every pair, the property is said to be incomplete, otherwise it is called complete or elusive. We show that the property of containing a complete subgraph with a given number of vertices is elusive. The proof also implies that the property of being r-chromatic (r fixed) is elusive.     

Large induced subgraphs with equated maximum degree

For a graph G, denote by f_k(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 , where n is the number of vertices of G. It is conjectured that for every fixed k. We prove this for k=2,3. While the proof for the case k=2 is easy, already the proof for the case k=3 is considerably more difficult. The case k=4 remains open.A related parameter, s_k(G), denotes the maximum integer m so that there are k vertex-disjoint subgraphs of G, each with m vertices, and with the same maximum degree. We prove that for every fixed k, s_k(G)≥n/k−o(n). The proof relies on probabilistic arguments.     

Heavy cycles in k-connected weighted graphs with large weighted degree sums

A weighted graph is one in which every edge e is assigned a nonnegative number w(e), called the weight of e. The weight of a cycle is defined as the sum of the weights of its edges. The weighted degree of a vertex is the sum of the weights of the edges incident with it. In this paper, we prove that: Let G be a k-connected weighted graph with k?2. Then G contains either a Hamilton cycle or a cycle of weight at least 2m/(k+1), if G satisfies the following conditions: (1) The weighted degree sum of any k+1 pairwise nonadjacent vertices is at least m; (2) In each induced claw and each induced modified claw of G, all edges have the same weight. This generalizes an early result of Enomoto et al. on the existence of heavy cycles in k-connected weighted graphs.     

Relative length of long paths and cycles in graphs with large degree sums

For a graph G, p(G) denotes the order of a longest path in G and c(G) the order of a longest cycle. We show that if G is a connected graph n ≥ 3 vertices such that d(u) + d(v) + d(w) ≧ n for all triples u, v, w of independent vertices, then G satisfies c(G) ≥ p(G) – 1, or G is in one of six families of exceptional graphs. This generalizes results of Bondy and of Bauer, Morgana, Schmeichel, and Veldman. © 1995, John Wiley & Sons, Inc.     

Graphs with unavoidable subgraphs with large degrees

Let ??(n, m) denote the class of simple graphs on n vertices and m edges and let G ∈ ?? (n, m). There are many results in graph theory giving conditions under which G contains certain types of subgraphs, such as cycles of given lengths, complete graphs, etc. For example, Turan's theorem gives a sufficient condition for G to contain a K_{k + 1} in terms of the number of edges in G. In this paper we prove that, for m = αn², α > (k - 1)/2k, G contains a K_{k + 1}, each vertex of which has degree at least f(α)n and determine the best possible f(α). For m = ?n²/4? + 1 we establish that G contains cycles whose vertices have certain minimum degrees. Further, for m = αn², α > 0 we establish that G contains a subgraph H with δ(H) ≥ f(α, n) and determine the best possible value of f(α, n).     

Complete rational arithmetic sums

Let {itq} > 1 be an integer number, \(f\left( x \right) = {a_n}{x^n} + \ldots + {a_1}x + {a_0}\) be a polynomial with integer coefficients, and ({ita}{in{itn}}, . . . ,{ita}{in1},{itq}) = 1. The following estimate is valid: \(\left| {S\left( {\frac{{f\left( x \right)}}{q}} \right)} \right| = \left| {\sum\limits_{x = 1}^q \rho \left( {\frac{{f\left( x \right)}}{q}} \right)} \right| \ll {q^{1 - 1/n}}\), where \(\rho \left( t \right) = 0,5 - \left\{ t \right\}\).     

On graphs with subgraphs having large independence numbers

Let G be a graph on n vertices in which every induced subgraph on vertices has an independent set of size at least . What is the largest so that every such G must contain an independent set of size at least q? This is one of the several related questions raised by Erd?s and Hajnal. We show that , investigate the more general problem obtained by changing the parameters s and t, and discuss the connection to a related Ramsey‐type problem. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 149–157, 2007     

Finding maximum subgraphs with relatively large vertex connectivity

We consider a clique relaxation model based on the concept of relative vertex connectivity. It extends the classical definition of a k-vertex-connected subgraph by requiring that the minimum number of vertices whose removal results in a disconnected (or a trivial) graph is proportional to the size of this subgraph, rather than fixed at k. Consequently, we further generalize the proposed approach to require vertex-connectivity of a subgraph to be some function f of its size. We discuss connections of the proposed models with other clique relaxation ideas from the literature and demonstrate that our generalized framework, referred to as f-vertex-connectivity, encompasses other known vertex-connectivity-based models, such as s-bundle and k-block. We study related computational complexity issues and show that finding maximum subgraphs with relatively large vertex connectivity is NP-hard. An interesting special case that extends the R-robust 2-club model recently introduced in the literature, is also considered. In terms of solution techniques, we first develop general linear mixed integer programming (MIP) formulations. Then we describe an effective exact algorithm that iteratively solves a series of simpler MIPs, along with some enhancements, in order to obtain an optimal solution for the original problem. Finally, we perform computational experiments on several classes of random and real-life networks to demonstrate performance of the developed solution approaches and illustrate some properties of the proposed clique relaxation models.     

Stability for large forbidden subgraphs

In this note we strengthen the stability theorem of Erd?s and Simonovits. Write K_r(s₁, …, s_r) for the complete r‐partite graph with classes of sizes s₁, …, s_r and T_r(n) for the r‐partite Turán graph of order n. Our main result is: For all r≥2 and all sufficiently small c>0, ε>0, every graph G of sufficiently large order n with e(G)>(1?1/r?ε)n²/2 satisfies one of the conditions:

• (a) G contains a $K_{r+1} (\lfloor c\,\mbox{ln}\,n \rfloor,\ldots,\lfloor c\,\mbox{ln}\,n \rfloor,\lceil n^{{1}-\sqrt{c}}\rceil)In this note we strengthen the stability theorem of Erd?s and Simonovits. Write K_r(s₁, …, s_r) for the complete r‐partite graph with classes of sizes s₁, …, s_r and T_r(n) for the r‐partite Turán graph of order n. Our main result is: For all r≥2 and all sufficiently small c>0, ε>0, every graph G of sufficiently large order n with e(G)>(1?1/r?ε)n²/2 satisfies one of the conditions:
  • (a) G contains a $K_{r+1} (\lfloor c\,\mbox{ln}\,n \rfloor,\ldots,\lfloor c\,\mbox{ln}\,n \rfloor,\lceil n^{{1}-\sqrt{c}}\rceil)$;
  • (b) G differs from T_r(n) in fewer than (ε^1/3+c^1/(3r+3))n² edges.
  Letting µ(G) be the spectral radius of G, we prove also a spectral stability theorem: For all r≥2 and all sufficiently small c>0, ε>0, every graph G of sufficiently large order n with µ(G)>(1?1/r?ε)n satisfies one of the conditions:
  • (a) G contains a $K_{r+1}(\lfloor c\,\mbox{ln}\,n\rfloor,\ldots,\lfloor c\,\mbox{ln}\,n\rfloor,\lceil n^{1-\sqrt{c}}\rceil)$;
  • (b) G differs from T_r(n) in fewer than (ε^1/4+c^1/(8r+8))n² edges.
  © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 362–368, 2009     

Largest bipartite subgraphs in triangle-free graphs with maximum degree three

Let G be a triangle-free, loopless graph with maximum degree three. We display a polynomial algorithm which returns a bipartite subgraph of G containing at least ? of the edges of G. Furthermore, we characterize the dodecahedron and the Petersen graph as the only 3-regular, triangle-free, loopless, connected graphs for which no bipartite subgraph has more than this proportion.     

A degree condition for spanning eulerian subgraphs

Let p ≥ 2 be a fixed integer. Let G be a simple and 2-edge-connected graph on n vertices, and let g be the girth of G. If d(u) + d(v) ≥ (2/(g ? 2))((n/p) ? 4 + g) holds whenever uv ? E(G), and if n is sufficiently large compared to p, then either G has a spanning eulerian subgraph or G can be contracted to a graph G₁ of order at most p without a spanning eulerian subgraph. Furthermore, we characterize the graphs that satisfy the conditions above such that G₁ has order p and does not have any spanning eulerian subgraph. © 1993 John Wiley & Sons, Inc.     

On the character degree sums

Let G be a finite group, and T(G) be the sum of all complex irreducible character degrees of G. In this paper, we get the exact lower bound of |G|∕T(G) for a non-r-solvable group G.     

Covering cycles andk-term degree sums

We show that if , for every stable set , then the vertex set ofG can be covered withk–1 cycles, edges or vertices. This settles a conjecture by Enomoto, Kaneko and Tuza.     

Graphs with many r-cliques have large complete r-partite subgraphs

Let r2 and c>0. Every graph on n vertices with at least cn^rcliques on r vertices contains a complete r-partite subgraphwith r–1 parts of size c^rlog n and one part of size greaterthan n^1–cr–1. This result implies a quantitativeform of the Erdös–Stone theorem.     

Complete subgraphs of the graphs of convex polytopes

It is shown that if three vertices of the graph G(l') of a convex 3-polytope P are chosen, then G(P) contains a refinement of the complete graph C₄ on four vertices, for which the three chosen vertices are principal (that is, correspond to vertices of C₄ in the refinement). In general, all four vertices may not be preassigned as principal. For dimensions d?4, simple (simplicial) d-polytopes are constructed whose graphs contain sets of three (four) vertices, which cannot all be principal in any refinement of C₄₊₁.