Minimum Degree Condition Research Articles

A minimum degree condition forcing complete graph immersion

An immersion of a graph H into a graph G is a one-to-one mapping f: V (H) → V (G) and a collection of edge-disjoint paths in G, one for each edge of H, such that the path P uv corresponding to edge uv has endpoints f(u) and f(v). The immersion is strong if the paths P uv are internally disjoint from f(V (H)). It is proved that for every positive integer Ht, every simple graph of minimum degree at least 200t contains a strong immersion of the complete graph K t . For dense graphs one can say even more. If the graph has order n and has 2cn 2 edges, then there is a strong immersion of the complete graph on at least c 2 n vertices in G in which each path P uv is of length 2. As an application of these results, we resolve a problem raised by Paul Seymour by proving that the line graph of every simple graph with average degree d has a clique minor of order at least cd 3/2, where c>0 is an absolute constant. For small values of t, 1≤t≤7, every simple graph of minimum degree at least t−1 contains an immersion of K t (Lescure and Meyniel [13], DeVos et al. [6]). We provide a general class of examples showing that this does not hold when t is large.

Minimum degree conditions for vertex-disjoint even cycles in large graphs

We prove a variant of a theorem of Corrádi and Hajnal (1963) [4] which says that if a graph G has at least 3k vertices and its minimum degree is at least 2k, then G contains k vertex-disjoint cycles. Specifically, our main result is the following. For any positive integer k, there is a constant ck such that if G is a graph with at least ck vertices and the minimum degree of G is at least 2k, then (i) G contains k vertex-disjoint even cycles, or (ii) (2k−1)K1∨pK2⊂G⊂K2k−1∨pK2 (p⩾k⩾2), or (iii) k=1 and each block of G is either a K2 or an odd cycle.

The maximum number of complete subgraphs in a graph with given maximum degree

Extremal problems involving the enumeration of graph substructures have a long history in graph theory. For example, the number of independent sets in a d-regular graph on n vertices is at most (2d+1−1)n/2d by the Kahn–Zhao theorem [7,13]. Relaxing the regularity constraint to a minimum degree condition, Galvin [5] conjectured that, for n⩾2d, the number of independent sets in a graph with δ(G)⩾d is at most that in Kd,n−d.In this paper, we give a lower bound on the number of independent sets in a d-regular graph mirroring the upper bound in the Kahn–Zhao theorem. The main result of this paper is a proof of a strengthened form of Galvinʼs conjecture, covering the case n⩽2d as well. We find it convenient to address this problem from the perspective of G¯. From this perspective, we show that the number of complete subgraphs of a graph G on n vertices with Δ(G)⩽r, where n=a(r+1)+b with 0⩽b⩽r, is bounded above by the number of complete subgraphs in aKr+1∪Kb.

Minimum vertex degree conditions for loose Hamilton cycles in 3-uniform hypergraphs

We investigate minimum vertex degree conditions for 3-uniform hypergraphs which ensure the existence of loose Hamilton cycles. A loose Hamilton cycle is a spanning cycle in which only consecutive edges intersect and these intersections consist of precisely one vertex.We prove that every 3-uniform n-vertex (n even) hypergraph H with minimum vertex degree δ1(H)⩾(716+o(1))(n2) contains a loose Hamilton cycle. This bound is asymptotically best possible.

Semi-degree threshold for anti-directed Hamiltonian cycles

In 1980, Grant initiated the study of minimum degree conditions for a directed graph D to contain an anti-directed Hamiltonian cycle (an orientation in which consecutive edges alternate direction). We prove that for sufficiently large even n, if D is a directed graph on n vertices with minimum out-degree and in-degree at least n2+1, then D contains an anti-directed Hamiltonian cycle. This result is sharp.

Minimum degree and disjoint cycles in generalized claw-free graphs

For s≥3 a graph is K1,s-free if it does not contain an induced subgraph isomorphic to K1,s. Cycles in K1,3-free graphs, called claw-free graphs, have been well studied. In this paper we extend results on disjoint cycles in claw-free graphs satisfying certain minimum degree conditions to K1,s-free graphs, normally called generalized claw-free graphs. In particular, we prove that if G is K1,s-free of sufficiently large order n=3k with δ(G)≥n/2+c for some constant c=c(s), then G contains k disjoint triangles. Analogous results with the complete graph K3 replaced by a complete graph Km for m≥3 will be proved. Also, the existence of 2-factors for K1,s-free graphs with minimum degree conditions will be shown.

A note on powers of Hamilton cycles in generalized claw-free graphs

Seymour conjectured for a fixed integer k≥2 that if G is a graph of order n with δ(G)≥kn/(k+1), then G contains the kth power Cnk of a Hamiltonian cycle Cn of G, and this minimum degree condition is sharp. Earlier the k=2 case was conjectured by Pósa. This was verified by Komlós et al. [4]. For s≥3, a graph is K1,s-free if it does not contain an induced subgraph isomorphic to K1,s. Such graphs will be referred to as generalized claw-free graphs. Minimum degree conditions that imply that a generalized claw-free graph G of sufficiently large order n contains a kth power of a Hamiltonian cycle will be proved. More specifically, it will be shown that for any ϵ>0 and for n sufficiently large, any K1,s-free graph of order n with δ(G)≥(1/2+ϵ)n contains a Cnk.

Locating pairs of vertices on Hamiltonian cycles

Given a fixed positive integer k≥2 and a fixed pair of vertices x and y in a graph of sufficiently large order n=n(k), minimum degree conditions that imply the existence of a Hamiltonian cycle C such that the distance on the cycle C between x and y is precisely k will be proved.

Minimum Degree and Disjoint Cycles in Claw-Free Graphs

A graph is claw-free if it does not contain an induced subgraph isomorphic to K1,3. Cycles in claw-free graphs have been well studied. In this paper we extend results on disjoint cycles in claw-free graphs satisfying certain minimum degree conditions. In particular, we prove that if G is claw-free of sufficiently large order n = 3k with δ(G) ≥ n/2, then G contains k disjoint triangles.

Minimum vertex degree conditions for loose Hamilton cycles in 3-uniform hypergraphs

Abstract We investigate minimum vertex degree conditions for 3-uniform hypergraphs which ensure the existence of loose Hamilton cycles. A loose Hamilton cycle is a spanning cycle in which consecutive edges intersect in a single vertex. We prove that every 3-uniform n-vertex (n even) hypergraph H with minimum vertex degree δ 1 ( H ) ⩾ ( 7 16 + o ( 1 ) ) ( n 2 ) contains a loose Hamilton cycle. This bound is asymptotically best possible.

Filling the gap between Turán's theorem and Pósa's conjecture

Much of extremal graph theory has concentrated either on finding very small subgraphs of a large graph (Turán-type results) or on finding spanning subgraphs (Dirac-type results). In this paper, we are interested in finding intermediate-sized subgraphs. We investigate minimum degree conditions under which a graph G contains squared paths and squared cycles of arbitrary specified lengths. We determine precise thresholds, assuming that the order of G is large. This extends results of Fan and Kierstead [J. Combin. Theory Ser. B 63 (1995) 55–64] and of Komlós, Sarközy and Szemerédi [Random Structures Algorithms 9 (1996) 193–211] concerning the containment of a spanning squared path and a spanning squared cycle, respectively. Our results show that such minimum degree conditions constitute not merely an interpolation between the corresponding Turán-type and Dirac-type results, but exhibit other interesting phenomena.

Distributing vertices on Hamiltonian cycles

Let G be a graph of order n and 3≤t≤n/4 be an integer. Recently, Kaneko and Yoshimoto [J Combin Theory Ser B 81(1) (2001), 100–109] provided a sharp δ(G) condition such that for any set X of t vertices, G contains a hamiltonian cycle H so that the distance along H between any two vertices of X is at least n/2t. In this article, minimum degree and connectivity conditions are determined such that for any graph G of sufficiently large order n and for any set of t vertices X⊆V(G), there is a hamiltonian cycle H so that the distance along H between any two consecutive vertices of X is approximately n/t. Furthermore, the minimum degree threshold is determined for the existence of a hamiltonian cycle H such that the vertices of X appear in a prescribed order at approximately predetermined distances along H. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 28–45, 2012

A New Bound on the Domination Number of Graphs with Minimum Degree Two

For a graph $G$, let $\gamma(G)$ denote the domination number of $G$ and let $\delta(G)$ denote the minimum degree among the vertices of $G$. A vertex $x$ is called a bad-cut-vertex of $G$ if $G-x$ contains a component, $C_x$, which is an induced $4$-cycle and $x$ is adjacent to at least one but at most three vertices on $C_x$. A cycle $C$ is called a special-cycle if $C$ is a $5$-cycle in $G$ such that if $u$ and $v$ are consecutive vertices on $C$, then at least one of $u$ and $v$ has degree $2$ in $G$. We let ${\rm bc}(G)$ denote the number of bad-cut-vertices in $G$, and ${\rm sc}(G)$ the maximum number of vertex disjoint special-cycles in $G$ that contain no bad-cut-vertices. We say that a graph is $(C_4,C_5)$-free if it has no induced $4$-cycle or $5$-cycle. Bruce Reed [Paths, stars and the number three. Combin. Probab. Comput. 5 (1996), 277–295] showed that if $G$ is a graph of order $n$ with $\delta(G) \ge 3$, then $\gamma(G) \le 3n/8$. In this paper, we relax the minimum degree condition from three to two. Let $G$ be a connected graph of order $n \ge 14$ with $\delta(G) \ge 2$. As an application of Reed's result, we show that $\gamma(G) \le \frac{1}{8} ( 3n + {\rm sc}(G) + {\rm bc}(G))$. As a consequence of this result, we have that (i) $\gamma(G) \le 2n/5$; (ii) if $G$ contains no special-cycle and no bad-cut-vertex, then $\gamma(G) \le 3n/8$; (iii) if $G$ is $(C_4,C_5)$-free, then $\gamma(G) \le 3n/8$; (iv) if $G$ is $2$-connected and $d_G(u) + d_G(v) \ge 5$ for every two adjacent vertices $u$ and $v$, then $\gamma(G) \le 3n/8$. All bounds are sharp.

A Note on Bipartite Graph Tiling

Bipartite graph tiling was studied by Zhao [SIAM J. Discrete Math., 23 (2009), pp. 888-900], who gave the best possible minimum degree conditions for a balanced bipartite graph on $2ms$ vertices to contain $m$ vertex disjoint copies of $K_{s,s}$. Let $s 2s+1$. We give the best possible minimum degree condition in this case.

The existence of a 2-factor in K1, n-free graphs with large connectivity and large edge-connectivity

In this article, we study the existence of a 2-factor in a K1, n-free graph. Sumner [J London Math Soc 13 (1976), 351–359] proved that for n⩾4, an (n−1)-connected K1, n-free graph of even order has a 1-factor. On the other hand, for every pair of integers m and n with m⩾n⩾4, there exist infinitely many (n−2)-connected K1, n-free graphs of even order and minimum degree at least m which have no 1-factor. This implies that the connectivity condition of Sumner's result is sharp, and we cannot guarantee the existence of a 1-factor by imposing a large minimum degree. On the other hand, Ota and Tokuda [J Graph Theory 22 (1996), 59–64] proved that for n⩾3, every K1, n-free graph of minimum degree at least 2n−2 has a 2-factor, regardless of its connectivity. They also gave examples showing that their minimum degree condition is sharp. But all of them have bridges. These suggest that the effects of connectivity, edge-connectivity and minimum degree to the existence of a 2-factor in a K1, n-free graph are more complicated than those to the existence of a 1-factor. In this article, we clarify these effects by giving sharp minimum degree conditions for a K1, n-free graph with a given connectivity or edge-connectivity to have a 2-factor. Copyright © 2010 Wiley Periodicals, Inc. J Graph Theory 68:77-89, 2011 © 2011 Wiley Periodicals, Inc.

Minimum degree and density of binary sequences

For d , k ∈ N with k ≤ 2 d , let g ( d , k ) denote the infimum density of binary sequences ( x i ) i ∈ Z ∈ { 0 , 1 } Z which satisfy the minimum degree condition ∑ j = 1 d ( x i + j + x i − j ) ≥ k for all i ∈ Z with x i = 1 . We reduce the problem of computing g ( d , k ) to a combinatorial problem related to the generalized k -girth of a graph G which is defined as the minimum order of an induced subgraph of G of minimum degree at least k . Extending results of Kézdy and Markert, and of Bermond and Peyrat, we present a minimum mean cycle formulation that yields g ( d , k ) for small values of d and k . For odd values of k with d + 1 ≤ k ≤ 2 d , we conjecture g ( d , k ) = k 2 − 1 2 ( d k − 1 ) and show that this holds for k ≥ 2 d − 3 .

Note on Bipartite Graph Tilings

Let $s<t$ be two fixed positive integers. We study sufficient minimum-degree conditions for a bipartite graph G, with both color classes of size $n=k(s+t)$, which ensure that G has a $K_{s,t}$-factor. Our result extends the work of Zhao, who determined the minimum-degree threshold which guarantees that a bipartite graph has a $K_{s,s}$-factor.

A minimum degree condition of fractional [formula omitted]-deleted graphs

Let G be a graph of order n, and let k ⩾ 1 and m ⩾ 1 be two integers. In this paper, we consider the relationship between the minimum degree δ ( G ) and the fractional ( k , m ) -deleted graphs. It is proved that if n ⩾ 4 k − 5 + 2 ( 2 k + 1 ) m and δ ( G ) ⩾ n 2 , then G is a fractional ( k , m ) -deleted graph. Furthermore, we show that the minimum degree condition is sharp in some sense. To cite this article: S. Zhou, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Cycles of given length in oriented graphs

We show that for each ℓ ⩾ 4 every sufficiently large oriented graph G with δ + ( G ) , δ − ( G ) ⩾ ⌊ | G | / 3 ⌋ + 1 contains an ℓ-cycle. This is best possible for all those ℓ ⩾ 4 which are not divisible by 3. Surprisingly, for some other values of ℓ, an ℓ-cycle is forced by a much weaker minimum degree condition. We propose and discuss a conjecture regarding the precise minimum degree which forces an ℓ-cycle (with ℓ ⩾ 4 divisible by 3) in an oriented graph. We also give an application of our results to pancyclicity and consider ℓ-cycles in general digraphs.

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 Turan's theorem [Turan, 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 Komlos, Sarkozy and Szemeredi [Komlos, J., G. N. Sarkozy and E. Szemeredi, On the square of a Hamiltonian cycle in dense graphs, Random Struct. Algorithms 9 (1996), 193-211; Komlos, J., G. N. Sarkozy and E. Szemeredi, Proof of the Seymour Conjecture for large graphs, Ann. Comb. 2 (1998), 43–60] towards a proof of the Posa-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.