A new characterization of matrices with the consecutive ones property

We consider the following constraint satisfaction problem: Given a set F of subsets of a finite set S of cardinality n, and an assignment of intervals of the discrete set {1,…,n} to each of the subsets, does there exist a bijection f:S→{1,…,n} such that for each element of F, its image under f is same as the interval assigned to it. An interval assignment to a given set of subsets is called feasible if there exists such a bijection. In this paper, we characterize feasible interval assignments to a given set of subsets. We then use this result to characterize matrices with the Consecutive Ones Property (COP), and to characterize matrices for which there is a permutation of the rows such that the columns are all sorted in ascending order. We also present a characterization of set systems which have a feasible interval assignment.     

The weighted consecutive ones problem for a fixed number of rows or columns

The NP-hard weighted consecutive ones problem consists of converting a given 0/1-matrix to a consecutive ones matrix by switching entries with a minimum total cost. Here we consider this problem when the number of rows or columns is fixed. We show that in this case the problem can be solved in polynomial time.     

Set covering with almost consecutive ones property

In this paper, we consider set covering problems with a coefficient matrix almost having the consecutive ones property, i.e., in most rows of the coefficient matrix, the ones appear consecutively and only a few blocks of consecutive ones appear in the remaining rows. If this property holds for all rows it is well known that the set covering problem can be solved efficiently. For our case of almost consecutive ones we present a reformulation exploiting the consecutive ones structure to develop bounds and a branching scheme. Our approach has been tested on real-world data as well as on theoretical problem instances.     

Roman domination in regular graphs

A Roman domination function on a graph G=(V(G),E(G)) is a function f:V(G)→{0,1,2} satisfying the condition that every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2. The weight of a Roman dominating function is the value f(V(G))=∑_u∈V(G)f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. Cockayne et al. [E. J. Cockayne et al. Roman domination in graphs, Discrete Mathematics 278 (2004) 11-22] showed that γ(G)≤γ_R(G)≤2γ(G) and defined a graph G to be Roman if γ_R(G)=2γ(G). In this article, the authors gave several classes of Roman graphs: P_3k,P_3k+2,C_3k,C_3k+2 for k≥1, K_m,n for min{m,n}≠2, and any graph G with γ(G)=1; In this paper, we research on regular Roman graphs and prove that: (1) the circulant graphs and , n⁄≡1 (mod (2k+1)), (n≠2k) are Roman graphs, (2) the generalized Petersen graphs P(n,2k+1)( (mod 4) and ), P(n,1) (n⁄≡2 (mod 4)), P(n,3) ( (mod 4)) and P(11,3) are Roman graphs, and (3) the Cartesian product graphs are Roman graphs.     

Eigenvalues and edge-connectivity of regular graphs

In this paper, we show that if the second largest eigenvalue of a d-regular graph is less than , then the graph is k-edge-connected. When k is 2 or 3, we prove stronger results. Let ρ(d) denote the largest root of x³-(d-3)x²-(3d-2)x-2=0. We show that if the second largest eigenvalue of a d-regular graph G is less than ρ(d), then G is 2-edge-connected and we prove that if the second largest eigenvalue of G is less than , then G is 3-edge-connected.     

Efficient edge domination in regular graphs

An induced matching of a graph G is a matching having no two edges joined by an edge. An efficient edge dominating set of G is an induced matching M such that every other edge of G is adjacent to some edge in M. We relate maximum induced matchings and efficient edge dominating sets, showing that efficient edge dominating sets are maximum induced matchings, and that maximum induced matchings on regular graphs with efficient edge dominating sets are efficient edge dominating sets. A necessary condition for the existence of efficient edge dominating sets in terms of spectra of graphs is established. We also prove that, for arbitrary fixed p≥3, deciding on the existence of efficient edge dominating sets on p-regular graphs is NP-complete.     

Size Ramsey numbers for some regular graphs

P. Erdös, R.J. Faudree, C.C. Rousseau and R.H. Schelp [P. Erdös, R.J. Faudree, C.C. Rousseau, R.H. Schelp, The size Ramsey number, Period. Math. Hungar. 9 (1978) 145-161] studied the asymptotic behaviour of for certain graphs G,H. In this paper there will be given a lower bound for the diagonal size Ramsey number of K_n,n,n. The result is a generalization of a theorem for K_n,n given by P. Erdös and C.C. Rousseau [P. Erdös, C.C. Rousseau, The size Ramsey numbers of a complete bipartite graph, Discrete Math. 113 (1993) 259-262].Moreover, an open question for bounds for size Ramsey number of each n-regular graph of order n+t for t>n−1 is posed.     

Hamiltonicity and pancyclicity of cartesian products of graphs

The cartesian product of a graph G with K₂ is called a prism over G. We extend known conditions for hamiltonicity and pancyclicity of the prism over a graph G to the cartesian product of G with paths, cycles, cliques and general graphs. In particular we give results involving cubic graphs and almost claw-free graphs.We also prove the following: Let G and H be two connected graphs. Let both G and H have a 2-factor. If Δ(G)≤g^′(H) and Δ(H)≤g^′(G) (we denote by g^′(F) the length of a shortest cycle in a 2-factor of a graph F taken over all 2-factorization of F), then G□H is hamiltonian.     

Dominating and large induced trees in regular graphs

Recently Chen et al. [Tree domination in graphs, Ars Combin. 73 (2004) 193-203] asked for characterizations of the class of graphs and the class of regular graphs that have an induced dominating tree, i.e. for which there exists a dominating set that induces a tree.We give a somewhat negative answer to their question by proving that the corresponding decision problems are NP-complete. Furthermore, we prove essentially best-possible lower bounds on the maximum order of induced trees in connected cacti of maximum degree 3 and connected cubic graphs.Finally, we give a forbidden induced subgraph condition for the existence of induced dominating trees.     

A note on a conjecture on maximum matching in almost regular graphs

Mkrtchyan, Petrosyan, and Vardanyan made the following conjecture: Every graph G with Δ(G)−δ(G)≤1 has a maximum matching whose unsaturated vertices do not have a common neighbor. We disprove this conjecture.     

The spectrum of infinite regular line graphs

Let be an infinite -regular graph and its line graph. We consider discrete Laplacians on and , and show the exact relation between the spectrum of and that of . Our method is also applicable to -semiregular graphs, subdivision graphs and para-line graphs.

     

Clustering of spectra and fractals of regular graphs

We exhibit a characteristic structure of the class of all regular graphs of degree d that stems from the spectra of their adjacency matrices. The structure has a fractal threadlike appearance. Points with coordinates given by the mean and variance of the exponentials of graph eigenvalues cluster around a line segment that we call a filar. Zooming-in reveals that this cluster splits into smaller segments (filars) labeled by the number of triangles in graphs. Further zooming-in shows that the smaller filars split into subfilars labeled by the number of quadrangles in graphs, etc. We call this fractal structure, discovered in a numerical experiment, a multifilar structure. We also provide a mathematical explanation of this phenomenon based on the Ihara-Selberg trace formula, and compute the coordinates and slopes of all filars in terms of Bessel functions of the first kind.     

The existence of even regular factors of regular graphs on the number of cut edges

For any even integer k and any integer i, we prove that a （kr ＋i）-regular multigraph contains a k-factor if it contains no more than kr - 3k/2＋ i ＋ 2 cut edges, and this result is the best possible to guarantee the existence of k-factor in terms of the number of cut edges. We further give a characterization for k-factor free regular graphs.     

图的能量与哈密尔顿性

设G是一个无向简单图, A(G)为$G$的邻接矩阵. 用G的补图的特征值给出G包含哈密尔顿路、哈密尔顿圈以及哈密尔顿连通图的充分条件; 其次用二部图的拟补图的特征值给出二部图包含哈密尔顿圈的充分条件. 这些结果改进了一些已知的结果.