Let Gn,d be a random d-regular graph with n vertices, where d = o（n）. Given a fixed graph H, YH denotes the number of induced copies of H in Gn d In this paper, the authors determine the threshold of the event ＂YH ...Let Gn,d be a random d-regular graph with n vertices, where d = o（n）. Given a fixed graph H, YH denotes the number of induced copies of H in Gn d In this paper, the authors determine the threshold of the event ＂YH 〉 0＂, and also obtain the induced subgraph counts inside the threshold interval.展开更多
A graph is called claw-free if it contains no induced subgrapn lsomorpmc to K1,3. Matthews and Sumner proved that a 2-connected claw-free graph G is Hamiltonian if every vertex of it has degree at least （[V（G）I - 2...A graph is called claw-free if it contains no induced subgrapn lsomorpmc to K1,3. Matthews and Sumner proved that a 2-connected claw-free graph G is Hamiltonian if every vertex of it has degree at least （[V（G）I - 2）/3. At the workshop CSzC （Novy Smokovec, 1993）, Broersma conjectured the degree condition of this result can be restricted only to end-vertices of induced copies of N （the graph obtained from a triangle by adding three disjoint pendant edges）. Fujisawa and Yamashita showed that the degree condition of Matthews and Sumner can be restricted only to end-vertices of induced copies of Z1 （the graph obtained from a triangle by adding one pendant edge）. Our main result in this paper is a characterization of all graphs H such that a 2-connected claw-free graph G is Hamiltonian if eachend-vertex of every induced copy of H in G has degree at least IV（G）I/3 ＋ 1. This gives an affirmative solution of the conjecture of Broersma up to an additive constant.end-vertex of every induced copy of H in G has degree at least IV（G）I/3 ＋ 1. This gives an affirmative solution of the conjecture of Broersma up to an additive constant.展开更多
The definition of the ascending subgraph decomposition was given by Alavi. It has been conjectured that every graph of positive size has an ascending subgraph decomposition. In this paper it is proved that the regular...The definition of the ascending subgraph decomposition was given by Alavi. It has been conjectured that every graph of positive size has an ascending subgraph decomposition. In this paper it is proved that the regular graphs under some conditions do have an ascending subgraph decomposition.展开更多
Two new hereditary classes of P 5-free graphs where the stability number can be found in polynomial time are proposed.They generalize several known results.
The traditional game of cops and robbers is played on undirected graph. Recently, the same game played on directed graph is getting attention by more and more people. We knew that if we forbid some subgraph we can bou...The traditional game of cops and robbers is played on undirected graph. Recently, the same game played on directed graph is getting attention by more and more people. We knew that if we forbid some subgraph we can bound the cop number of the corresponding class of graphs. In this paper, we analyze the game of cops and robbers on H^(-)-free digraphs. However, it is not the same as the case of undirected graph. So we give a new concept(H^(-)^(*)-free digraph) to get a similar conclusion about the case of undirected graph.展开更多
基金This research is supported by the National Natural Science of Foundation under Grant Nos.10531070 and 10721101 of China.
文摘Let Gn,d be a random d-regular graph with n vertices, where d = o（n）. Given a fixed graph H, YH denotes the number of induced copies of H in Gn d In this paper, the authors determine the threshold of the event ＂YH 〉 0＂, and also obtain the induced subgraph counts inside the threshold interval.
基金Supported by NSFC(Grant Nos.11271300 and 11571135)the project NEXLIZ–CZ.1.07/2.3.00/30.0038+1 种基金the project P202/12/G061 of the Czech Science Foundation and by the European Regional Development Fund(ERDF)the project NTIS-New Technologies for Information Society,European Centre of Excellence,CZ.1.05/1.1.00/02.0090
文摘A graph is called claw-free if it contains no induced subgrapn lsomorpmc to K1,3. Matthews and Sumner proved that a 2-connected claw-free graph G is Hamiltonian if every vertex of it has degree at least （[V（G）I - 2）/3. At the workshop CSzC （Novy Smokovec, 1993）, Broersma conjectured the degree condition of this result can be restricted only to end-vertices of induced copies of N （the graph obtained from a triangle by adding three disjoint pendant edges）. Fujisawa and Yamashita showed that the degree condition of Matthews and Sumner can be restricted only to end-vertices of induced copies of Z1 （the graph obtained from a triangle by adding one pendant edge）. Our main result in this paper is a characterization of all graphs H such that a 2-connected claw-free graph G is Hamiltonian if eachend-vertex of every induced copy of H in G has degree at least IV（G）I/3 ＋ 1. This gives an affirmative solution of the conjecture of Broersma up to an additive constant.end-vertex of every induced copy of H in G has degree at least IV（G）I/3 ＋ 1. This gives an affirmative solution of the conjecture of Broersma up to an additive constant.
文摘The definition of the ascending subgraph decomposition was given by Alavi. It has been conjectured that every graph of positive size has an ascending subgraph decomposition. In this paper it is proved that the regular graphs under some conditions do have an ascending subgraph decomposition.
基金The first author was supported by DIMACS Summer2 0 0 3Award
文摘Two new hereditary classes of P 5-free graphs where the stability number can be found in polynomial time are proposed.They generalize several known results.
文摘The traditional game of cops and robbers is played on undirected graph. Recently, the same game played on directed graph is getting attention by more and more people. We knew that if we forbid some subgraph we can bound the cop number of the corresponding class of graphs. In this paper, we analyze the game of cops and robbers on H^(-)-free digraphs. However, it is not the same as the case of undirected graph. So we give a new concept(H^(-)^(*)-free digraph) to get a similar conclusion about the case of undirected graph.
