Degree Sum Condition Research Articles

A new condition on dominated pair degree sum for a digraph to be supereulerian

A new condition on dominated pair degree sum for a digraph to be supereulerian

Refinements of degree conditions for the existence of a spanning tree without small degree stems

Refinements of degree conditions for the existence of a spanning tree without small degree stems

Component factors and degree sum conditions in graphs

For a set A of connected graphs, an A -factor is a spanning subgraph of a graph, whose connected components are isomorphic to graphs from the set A . An A -factor is also referred as a component factor. A graph G is called an (A, m)-factor deleted graph if for every E0 ⊆ E(G) with |E0| = m,G−E0 admits an A -factor. A graph G is called an (A, l)-factor critical graph if for every V0 ⊆V(G) with |V0| = l,G−V0 admits an A -factor. Let m,l and k be three positive integers with k ≥ 2, and write F = {P2,C3,P5,T (3)} and H = {K1,1,K1,2,...,K1,k,T (2k+1)}, where T (3) and T (2k+1) are two special families of trees. Inspired by finding a sufficient condition to check for the existence of path-factors with some special restraints, we focus on the sufficient conditions based on a graphic parameter called degree sum: σk(G)=min_{X\subseteq V(G)}{[[EQUATION]]dG(x): X is an independent set of k vertices}. In this article, we verify that (i) an (l+2)-connected graph G of order n is an (\mathcal{F},l)-factor critical graph if σ3(G)\geq\frac{6n+9l}{5}; (ii) a (2m+1)-connected graph G of order n is an (\mathcal{F},m)-factor deleted graph if σm+2(G)\geq\frac{6}{5}n; (iii) an (l+2)-connected graph G of order n is an (\mathcal{H},l)-factor critical graph if σ2k+1(G)\geq\frac{6n+(6k+3)l}{2k+3}; (iv) a (2m+1)-connected graph G of order n is an (\mathcal{H},m)-factor deleted graph if σm+2(G)\geq\frac{6n}{2k+3}.

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.

On graph-based network parameters and component factors in networks

Many physical structures can conveniently be simulated by networks. To study the properties of the network, we use a graph to simulate the network. A graph H is called an F-factor of a graph G, if H is a spanning subgraph of G and every connected component of H is isomorphic to a graph from the graph set F. An F-factor is also referred as a component factor. The graph-based network parameter degree sum of G is defined by $\sigma_k(G)=\min_{X\subseteq V(G)}\{\sum_{x\in X}d_G(x): X is an independent set of k vertices\}.$ In this article, we give the precise degree sum condition for a graph to have $\{P_2,C_3, P_5, \mathcal{T}（3）\}$-factor and $\{K_{1,1}, K_{1,2},..., K_{1,k},\mathcal{T}(2k+1)\}$-factor. We also obtain similar results for $\{P_2,C_3, P_5, \mathcal{T}（3）\}$-factor avoidable graph and $\{K_{1,1}, K_{1,2},..., K_{1,k},\mathcal{T}(2k+1)\}$-factor avoidable graph, respectively.

Hamilton-connected claw-free graphs with Ore-degree conditions

Hamilton-connected claw-free graphs with Ore-degree conditions

A degree condition for cycles passing through specified vertices and edges

A degree condition for cycles passing through specified vertices and edges

Degree sum conditions for path-factor uniform graphs

Degree sum conditions for path-factor uniform graphs

Degree sum condition on distance 2 vertices for hamiltonian cycles in balanced bipartite graphs

Degree sum condition on distance 2 vertices for hamiltonian cycles in balanced bipartite graphs

Disjoint Cycles and Degree Sum Condition in a Graph

Disjoint Cycles and Degree Sum Condition in a Graph

Spectral radius and spanning trees of graphs

Spectral radius and spanning trees of graphs

Degree Conditions for Claw-Free Graphs to Have Spanning Trees with at Most Five Branch Vertices and Leaves in Total

A leaf of a tree is a vertex of degree one and a branch vertex of a tree is a vertex of degree at least three. In this paper, we show a degree condition for a claw-free graph to have spanning trees with at most five branch vertices and leaves in total. Moreover, the degree sum condition is best possible.

Degree sum condition for vertex-disjoint 5-cycles

Degree sum condition for vertex-disjoint 5-cycles

Dominated pair degree sum conditions of supereulerian digraphs

Dominated pair degree sum conditions of supereulerian digraphs

A Degree Sum Condition for Hamiltonian Graphs

A Degree Sum Condition for Hamiltonian Graphs

Degree sum conditions for hamiltonian index

In this note, we show a sharp lower bound of min left{{sumnolimits_{i = 1}^k {{d_G}({u_i}):{u_1}{u_2} ldots {u_k}}} right. is a path of (2-)connected G on its order such that (k-1)-iterated line graphs Lk−1(G) are hamiltonian.

Degree sum conditions for the existence of homeomorphically irreducible spanning trees

AbstractIn 1990, Albertson, Berman, Hutchinson, and Thomassen proved a theorem which gives a minimum degree condition for the existence of a spanning tree with no vertices of degree 2. Such a spanning tree is called a homeomorphically irreducible spanning tree (HIST). In this paper, we prove that every graph of order () contains a HIST if for any nonadjacent vertices and . The degree sum 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.

Degree sum condition for the existence of spanning k-trees in star-free graphs

Degree sum condition for the existence of spanning <i>k</i>-trees in star-free graphs

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.