Author
Hong-Jian Lai
Other affiliations: University of West Virginia, Wayne State University, Xinjiang University ...read more
Bio: Hong-Jian Lai is an academic researcher from West Virginia University. The author has contributed to research in topics: Line graph & Bound graph. The author has an hindex of 25, co-authored 274 publications receiving 2516 citations. Previous affiliations of Hong-Jian Lai include University of West Virginia & Wayne State University.
Topics: Line graph, Bound graph, Graph power, Cubic graph, Combinatorics
Papers published on a yearly basis
Papers
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TL;DR: This paper proves Catlin’s conjecture that if a connected graph G is at most two edges short of having two edge-disjoint spanning trees, then either G is supereulerian or G can be contracted to a K1 or a K2,t for some odd integer t 1.
90 citations
TL;DR: It is conjectured that every 3-edge-connected graph is A-connected, for every abelian group A with |A|≥5; and that every 5-edge
Abstract: Let A be a finite abelian group and G be a digraph. The boundary of a func- tion f : EOGU7!A is a function q f : VOGU7!A given by q fOvUa P e leaving v fOeUˇ P e entering v fOeU. The graph G is A-connected if for every b : VOGU7!A with P vA VOGU bOvUa0, there is a function f : EOGU7!Af0g such that q f a b. In (J. Combi- natorial Theory, Ser. B 56 (1992) 165-182), Jaeger et al showed that every 3-edge- connected graph is A-connected, for every abelian group A with jAjV 6. It is con- jectured that every 3-edge-connected graph is A-connected, for every abelian group A with jAjV 5; and that every 5-edge-connected graph is A-connected, for every abelian group A with jAjV 3. In this note, we investigate the group connectivity of 3-edge-connected chordal graphs and characterize 3-edge-connected chordal graphs that are A-connected for every finite abelian group A withjAjV 3.
87 citations
TL;DR: It is concluded that the hypercube structure is resilient as it includes a large connected component in the presence of large number of faulty vertices.
Abstract: Hypercube is one of the most popular topologies for connecting processors in multicomputer systems. In this paper we address the maximum order of a connected component in a faulty cube. The results established include several known conclusions as special cases. We conclude that the hypercube structure is resilient as it includes a large connected component in the presence of large number of faulty vertices.
86 citations
Journal Article•
TL;DR: The best possible upper bounds as an analogue to the Brook’s Theorem are proved, together with the determination of chromatic numbers for complete k-partite graphs.
Abstract: A proper vertex k-coloring of a graph G is dynamic if for every vertex v with degree at least 2, the neighbors of v receive at least two different colors. The smallest integer k such that G has a dynamic k-coloring is the dynamic chromatic number χd(G). We prove in this paper the following best possible upper bounds as an analogue to the Brook’s Theorem, together with the determination of chromatic numbers for complete k-partite graphs. (1) If ∆ ≤ 3, then χd(G) ≤ 4, with the only exception that G = C5, in which case χd(C5) = 5. (2) If ∆ ≥ 4, then χd(G) ≤ ∆+ 1. (3) χd(K1,1) = 2, χd(K1,m) = 3 and χd(Km,n) = 4 for m,n ≥ 2; χd(Kn1,n2,···,nk) = k for k ≥ 3.
81 citations
TL;DR: It is demonstrated that γ and η are closely related to the principal partition and can be used to give a simple definition of both the principal partitions and the more recent refinements of it.
70 citations
Cited by
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TL;DR: Some of the major results in random graphs and some of the more challenging open problems are reviewed, including those related to the WWW.
Abstract: We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.
7,116 citations
TL;DR: This paper analyzes the literature from the point of view of swarm engineering and proposes two taxonomies: in the first taxonomy, works that deal with design and analysis methods are classified; in the second, works according to the collective behavior studied are classified.
Abstract: Swarm robotics is an approach to collective robotics that takes inspiration from the self-organized behaviors of social animals. Through simple rules and local interactions, swarm robotics aims at designing robust, scalable, and flexible collective behaviors for the coordination of large numbers of robots. In this paper, we analyze the literature from the point of view of swarm engineering: we focus mainly on ideas and concepts that contribute to the advancement of swarm robotics as an engineering field and that could be relevant to tackle real-world applications. Swarm engineering is an emerging discipline that aims at defining systematic and well founded procedures for modeling, designing, realizing, verifying, validating, operating, and maintaining a swarm robotics system. We propose two taxonomies: in the first taxonomy, we classify works that deal with design and analysis methods; in the second taxonomy, we classify works according to the collective behavior studied. We conclude with a discussion of the current limits of swarm robotics as an engineering discipline and with suggestions for future research directions.
1,405 citations
Book•
17 Dec 1994
TL;DR: In this article, the Conjectures of Hadwiger and Hajos are used to define graph types, such as planar graph, graph on higher surfaces, and critical graph.
Abstract: Planar Graphs. Graphs on Higher Surfaces. Degrees. Critical Graphs. The Conjectures of Hadwiger and Hajos. Sparse Graphs. Perfect Graphs. Geometric and Combinatorial Graphs. Algorithms. Constructions. Edge Colorings. Orientations and Flows. Chromatic Polynomials. Hypergraphs. Infinite Chromatic Graphs. Miscellaneous Problems. Indexes.
1,380 citations
31 May 1993
TL;DR: Significant progress has been made with solution of location problems and in preprocessing and decomposition for discrete optimization and on the application of techniques from combinational optimization to nonlinear problems.
Abstract: : Significant progress has been made with solution of location problems and in preprocessing and decomposition for discrete optimization. There has also been research on the application of techniques from combinational optimization to nonlinear problems.
421 citations