Minimum Degree Sum Research Articles

On Vertex-Disjoint Chorded Cycles and Degree Sum Conditions

In this paper, we consider a degree sum condition sufficient to imply the existence of k vertex-disjoint chorded cycles in a graph G . Let σ 4 ( G ) be the minimum degree sum of four independent vertices of G . We prove that if G is a graph of order at least 11 k + 7 and σ 4 ( G ) ≥ 12 k − 3 with k ≥ 1 , then G contains k vertex-disjoint chorded cycles. We also show that the degree sum condition on σ 4 ( G ) is sharp.

A Spanning Tree with at Most $k$ Leaves in a $K_{1,p}$-Free Graph

For an integer $k \geq 2$, a tree is called a $k$-ended tree if it has at most $k$ leaves. It was shown that some $\sigma_{k+1}(G)$ conditions assure the existence of a spanning $k$-ended tree in a connected $K_{1,p}$-free graph $G$ for the pairs $(p,k)$ with $p \leq 4$, or $p= 5$ and $k=4,6$, where $\sigma_{k+1}(G)$ is the minimum degree sum of pairwise non-adjacent $k+1$ vertices of $G$. In this paper, we extend those results to the case with any integer $p \geq 3$ by proving that for any $k \geq 2$ and $p \geq 3$, there exists a constant $f(p,k)$ depending only on $k$ and $p$ such that if a connected $K_{1,p}$-free graph satisfies $\sigma_{k+1}(G) \ge |G|+ f(p,k)$, then $G$ has a spanning $k$-ended tree. The coefficient $1$ of $|G|$ in the $\sigma_{k+1}(G)$ condition is best possible.

On the Path Cover Number of Connected Quasi-Claw-Free Graphs

Detecting vertex disjoint paths is one of the central issues in designing and evaluating an interconnection network. It is naturally related to routing among nodes and fault tolerance of the network. A path cover of a graph G is a spanning subgraph of G consisting of vertex disjoint paths, and a path cover number of G denoted by p(G) = min{|P| : P is a path cover of G}. In this paper, we show that if the minimum degree sum of an independent set with k + 1 vertices in a connected quasi-claw-free graph G of order n is no less than n - k, then p(G) ≤ k - 1, where k ≥ 2. Examples illustrate that the degree sum condition in our result is sharp.

On Degree Sum Conditions and Vertex-Disjoint Chorded Cycles

In this paper, we consider a general degree sum condition sufficient to imply the existence of k vertex-disjoint chorded cycles in a graph G. Let $$\sigma _t(G)$$ be the minimum degree sum of t independent vertices of G. We prove that if G is a graph of sufficiently large order and $$\sigma _t(G)\ge 3kt-t+1$$ with $$k\ge 1$$ , then G contains k vertex-disjoint chorded cycles. We also show that the degree sum condition on $$\sigma _t(G)$$ is sharp. To do this, we also investigate graphs without chorded cycles.

Color degree sum conditions for properly colored spanning trees in edge-colored graphs

For a vertex v of an edge-colored graph, the color degree of v is the number of colors appeared in edges incident with v. An edge-colored graph is called properly colored if no two adjacent edges have the same color. In this paper, we prove that if the minimum color degree sum of two adjacent vertices of an edge-colored connected graph G is at least |G|, then G has a properly colored spanning tree. This is a generalization of the result proved by Cheng, Kano and Wang. We also show the sharpness of this lower bound of the color degree sum.

On sufficient degree conditions for traceability of claw-free graphs

We present new results on the traceability of claw-free graphs. In particular, we consider sufficient minimum degree and degree sum conditions that imply that these graphs admit a Hamilton path, unless they have a small order or they belong to well-defined classes of exceptional graphs. Our main result implies that a 2-connected claw-free graph G of sufficiently large order n with δ(G)≥3 is traceable if the degree sum of any set of t independent vertices of G is at least t(n+6)6, where t∈{1,2,…,6}, and that this lower bound n+66 on the degree sums is asymptotically sharp. Our results also imply that a 2-connected claw-free graph G of sufficiently large order n with minimum degree δ(G)≥22 is traceable if the degree sum of any set of t independent vertices of G is at least t(2n−5)14, where t∈{1,2,…,7}, unless G is a member of well-defined classes of exceptional graphs depending on t, and that this lower bound 2n−514 on the degree sums is asymptotically sharp. Our results also imply that a 2-connected claw-free graph G of sufficiently large order n with δ(G)≥18 is traceable if the degree sum of any set of 6 independent vertices is larger than n−6, and that this lower bound on the degree sums is sharp.

Long Cycles Passing Through a Linear Forest

A graph F is called a linear forest if $$V(F)=E(F)=\emptyset$$ or every component of F is a path. We denote by $$\omega _1(F)$$ the number of components of order 1 in F. In this article, we prove the following theorem. Let $$k\ge 5$$ and $$m\ge 0$$. Let G be a $$(k+m)$$-connected graph and F be a linear forest on a cycle of G with $$|E(F)|=m$$ and $$k + 1\le \omega _1(F) \le \lfloor \frac{4k-1}{3}\rfloor$$. Then G has a cycle of length at least $$\min \{\sigma _{2}(G)-m, |V(G)|\}$$ passing through F, where $$\sigma _{2}(G)$$ denotes the minimum degree sum of two independent vertices. Our result generalizes the theorem of Hu and Song (J Graph Theory 87(3):374–393).

Partitioning a graph into cycles with a specified number of chords

AbstractFor a graph , let be the minimum degree sum of two nonadjacent vertices in . A chord of a cycle in a graph is an edge of joining two nonconsecutive vertices of the cycle. In this paper, we prove the following result, which is an extension of a result of Brandt et al for large graphs: For positive integers and , there exists an integer such that if is a graph of order and , then can be partitioned into vertex‐disjoint cycles, each of which has at least chords.

A sharp Ore-type condition for a connected graph with no induced star to have a Hamiltonian path

We say a graph G has a Hamiltonian path if it has a path containing all vertices of G. For a graph G, let σ2(G) denote the minimum degree sum of two nonadjacent vertices of G; restrictions on σ2(G) are known as Ore-type conditions. Given an integer t≥5, we prove that if a connected graph G on n vertices satisfies σ2(G)>t−3t−2n, then G has either a Hamiltonian path or an induced subgraph isomorphic to K1,t. Moreover, we characterize all n-vertex graphs G where σ2(G)=t−3t−2n and G has neither a Hamiltonian path nor an induced subgraph isomorphic to K1,t. This is an analogue of a recent result by Momège (2018), who investigated the case when t=4.

Minimum Path Cover in Quasi-claw-Free Graphs

A path cover of a graph G is a spanning subgraph of G consisting of vertex disjoint paths; a minimum path cover of G is a path cover of G consisting of minimum number of paths; a path cover number of G, denoted by p(G), is the number of paths in a minimum path cover of G, i.e., $$p(G)=\min \{|{\mathcal {P}}|:$$$${\mathcal {P}}$$ is a path cover of $$G\}.$$ A graph G is quasi-claw-free if for any two vertices x, y with $$d(x,y)=2$$ in G, there exists a vertex u with $$u\in N(x)\cap N(y)$$ and $$N(u)\subseteq N[x]\cup N[y].$$ In this paper, we prove that for any quasi-claw-free graph G of order n, if $$\sigma _{k+1}(G)\ge {n-k}$$ for a positive integer k, then $$p(G)\le k,$$ where $$\sigma _{k+1}(G)$$ is the minimum degree sum of an independent set with $$k+1$$ vertices in G. Our result is a generalization of Ore’s sufficient conditions of hamiltonicity for quasi-claw-free graphs.

Spanning Trees with At Most 6 Leaves in K1,5-Free Graphs

A graph G is called K1,5-free if G contains no K1,5 as an induced subgraph. A tree with at most m leaves is called an m-ended tree. Let σkG be the minimum degree sum of k independent vertices in G. In this paper, it is shown that every connected K1,5-free graph G contains a spanning 6-ended tree if σ7G≥G−2.

Spanning 5-Ended Trees in $$K_{1,5}$$-Free Graphs

A graph G is called $$K_{1,5}$$ -free if G contains no $$K_{1,5}$$ as an induced subgraph. A tree with at most m leaves is called an m-ended tree. Let $$\sigma _k(G)$$ denote the minimum degree sum of k independent vertices in G. In this paper, it is shown that every connected $$K_{1,5}$$ -free graph G contains a spanning 5-ended tree if $$\sigma _6(G)\ge |G|-1$$ .

Degree sum and graph linkage with prescribed path lengths

Given a fixed graph H, we provide a sharp degree-sum condition such that if a graph G is sufficiently large with sufficiently large minimum degree sum, then it contains a spanning H-subdivision in which the ground vertices and lengths of the corresponding edge paths are specified.

Vertex Degree Sums for Perfect Matchings in 3-Uniform Hypergraphs

We determine the minimum degree sum of two adjacent vertices that ensures a perfect matching in a 3-uniform hypergraph without an isolated vertex. Suppose that $H$ is a 3-uniform hypergraph whose order $n$ is sufficiently large and divisible by $3$. If $H$ contains no isolated vertex and $\deg(u)+\deg(v) > \frac{2}{3}n^2-\frac{8}{3}n+2$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $H$ contains a perfect matching. This bound is tight and the (unique) extremal hyergraph is a different space barrier from the one for the corresponding Dirac problem.

Some Ore-type Results for Matching and Perfect Matching in k-uniform Hypergraphs

Let S1 and S2 be two (k − 1)-subsets in a k-uniform hypergraph H. We call S1 and S2 strongly or middle or weakly independent if H does not contain an edge e ∈ E(H) such that S1 ∩ e ≠ ∅ and S2 ∩ e ≠ ∅ or e ⊆ S1 ∪ S2 or e ⊇ S1 ∪ S2, respectively. In this paper, we obtain the following results concerning these three independence. (1) For any n ≥ 2k2 − k and k ≥ 3, there exists an n-vertex k-uniform hypergraph, which has degree sum of any two strongly independent (k − 1)-sets equal to 2n−4(k−1), contains no perfect matching; (2) Let d ≥ 1 be an integer and H be a k-uniform hypergraph of order n ≥ kd+(k−2)k. If the degree sum of any two middle independent (k−1)-subsets is larger than 2(d−1), then H contains a d-matching; (3) For all k ≥ 3 and sufficiently large n divisible by k, we completely determine the minimum degree sum of two weakly independent (k − 1)-subsets that ensures a perfect matching in a k-uniform hypergraph H of order n.

On the existence of vertex-disjoint subgraphs with high degree sum

For a graph G, we denote by σ2(G) the minimum degree sum of two non-adjacent vertices if G is non-complete; otherwise, σ2(G)=+∞. In this paper, we prove the following two results: (i) If s1,s2≥2 are integers and G is a non-complete graph with σ2(G)≥2(s1+s2+1)−1, then G contains two vertex-disjoint subgraphs H1 and H2 such that each Hi is a graph of order at least si+1 with σ2(Hi)≥2si−1. (ii) If s1,s2≥2 are integers and G is a triangle-free graph of order at least 3 with σ2(G)≥2(s1+s2)−1, then G contains two vertex-disjoint subgraphs H1 and H2 such that each Hi is a graph of order at least 2si with σ2(Hi)≥2si−1. By using this result, we also give some corollaries concerning degree conditions for the existence of k vertex-disjoint cycles.

A note on degree sum conditions for 2-factors with a prescribed number of cycles in bipartite graphs

Let G be a balanced bipartite graph of order 2n≥4, and let σ1,1(G) be the minimum degree sum of two non-adjacent vertices in different partite sets of G. In 1963, Moon and Moser proved that if σ1,1(G)≥n+1, then G is hamiltonian. In this note, we show that if k is a positive integer, then the Moon–Moser condition also implies the existence of a 2-factor with exactly k cycles for sufficiently large graphs. In order to prove this, we also give a σ1,1 condition for the existence of k vertex-disjoint alternating cycles with respect to a chosen perfect matching in G.

Decomposition methods based on articulation vertices for degree-dependent spanning tree problems

Decomposition methods for optimal spanning trees on graphs are explored in this work. The attention is focused on optimization problems where the objective function depends only on the degrees of the nodes of the tree. In particular, we deal with the Minimum Leaves problem, the Minimum Branch Vertices problem and the Minimum Degree Sum problem. The decomposition is carried out by identifying the articulation vertices of the graph and then its blocks, solving certain subproblems on the blocks and then bringing together the optimal sub-solutions following adequate procedures. Computational results obtained using similar Integer Programming formulations for both the original and the decomposed problems show the advantage of the proposed methods on decomposable graphs.

Placing Specified Vertices at Precise Locations on a Hamiltonian Cycle

Sharp minimum degree and degree sum conditions are proven for the existence of a Hamiltonian cycle passing through specified vertices with prescribed distances between them in large graphs.

Every Triangulated 3-Polytope of Minimum Degree 4 has a 4-Path of Weight at Most 27

By $\delta$ and $w_k$ denote the minimum degree and minimum degree-sum (weight) of a $k$-vertex path in a given graph, respectively. For every 3-polytope, $w_2\le13$ (Kotzig, 1955) and $w_3\le21$ (Ando, Iwasaki, Kaneko, 1993), where both bounds are sharp. For every 3-polytope with $\delta\ge4$, we have sharp bounds $w_2\le11$ (Lebesgue, 1940) and $w_3\le17$ (Borodin, 1997).Madaras (2000) proved that every triangulated 3-polytope with $\delta\ge4$ satisfies $w_4\le31$ and constructed such a 3-polytope with $w_4=27$.We improve the Madaras bound $w_4\le31$ to the sharp bound $w_4\le27$.