Cun-Quan Zhang

On (k, d)-colorings and fractional nowhere-zero flows

Let Sm denote the m-vertex simple digraph formed by m - 1 edges with a common tail. Let f(m) denote the minimum n such that every n-vertex tournament has a spanning subgraph consisting of n-m disjoint copies of Sm. We prove that m lg m - m lg lg m ≤ f(m) ≤ 4m2 - 6m for sufficiently large m.

Graphs with the circuit cover property

A circuit cover of an edge-weighted graph (G, p) is a multiset of circuits in G such that every edge e is contained in exactly p(e) circuits in the multiset. A nonnegative integer valued weight vector p is admissible if the total weight of any edge-cut is even, and no edge has more than half the total weight of any edge-cut containing it. A graph G has the circuit cover property if (G, p) has a circuit cover for every admissible weight vector p . We prove that a graph has the circuit cover property if and only if it contains no subgraph homeomorphic to Petersens graph. In particular, every 2-edge-connected graph with no subgraph homeomorphic to Petersens graph has a cycle double cover.

Emergence of segregation in evolving social networks

In many social networks, there is a high correlation between the similarity of actors and the existence of relationships between them. This paper introduces a model of network evolution where actors are assumed to have a small aversion from being connected to others who are dissimilar to themselves, and yet no actor strictly prefers a segregated network. This model is motivated by Schelling’s [Schelling TC (1969) Models of segregation. Am Econ Rev 59:488–493] classic model of residential segregation, and we show that Schelling’s results also apply to the structure of networks; namely, segregated networks always emerge regardless of the level of aversion. In addition, we prove analytically that attribute similarity among connected network actors always reaches a stationary distribution, and this distribution is independent of network topology and the level of aversion bias. This research provides a basis for more complex models of social interaction that are driven in part by the underlying attributes of network actors and helps advance our understanding of why dysfunctional social network structures may emerge.

Hamilton cycles in claw-free graphs

Bondy conjectured that if G is a k-connected graph of order n such that for any (k + 1)-independent set / of G, then the subgraph outside any longest cycle contains no path of length k − 1. In this paper, we are going to prove that, if G is a k-connected claw-free (K1,3-free) graph of order n such that for any (k + 1)-independent set /, then G contains a Hamilton cycle. The theorem in this paper implies Bondys conjecture in the case of claw-free graphs.

Finding critical independent sets and critical vertex subsets are polynomial problems

An independent set

(2 + ε)-Coloring of planar graphs with large odd-girth

J_c

A Novel Model for DNA Sequence Similarity Analysis Based on Graph Theory

of a graph G is called critical if \[ | J_c | - | N ( J_c ) | = \max \{ | J | - | N ( J ) |:J\,\text{is an independent set of }G \}, \] and a vertex subset

Optimal local community detection in social networks based on density drop of subgraphs

U_c

A new clustering method and its application in social networks

is called critical if \[ | U_c | - | N ( U_c ) | = \max \{ | U | - | N ( U ) |:U\,\text{is a vertex subset of }G \} . \] In this paper, it will be shown that finding a critical independent set and a critical vertex subset of a graph are solvable in polynomial time.

On flows in bidirected graphs

Let G be a 2-connected graph, let u and v be distinct vertices in V(G), and let X be a set of at most four vertices lying on a common (u, v)-path in G. If deg(x) ≥ d for all x e V(G) \ {u, v}, then there is a (u, v)-path P in G with X ‚ V(P) and |E(P)| ≥ d.