Clique-perfectness of complements of line graphs

The clique-transversal number τ_c(G) of a graph G is the minimum size of a set of vertices meeting all the cliques. The clique-independence number α_c(G) of G is the maximum size of a collection of vertex-disjoint cliques. A graph is clique-perfect if these two numbers are equal for every induced subgraph of G. Unlike perfect graphs, the class of clique-perfect graphs is not closed under graph complementation nor is a characterization by forbidden induced subgraphs known. Nevertheless, partial results in this direction have been obtained. For instance, in [Bonomo, F., M. Chudnovsky and G. Durán, Partial characterizations of clique-perfect graphs I: Subclasses of claw-free graphs, Discrete Appl. Math. 156 (2008), pp. 1058–1082], a characterization of those line graphs that are clique-perfect is given in terms of minimal forbidden induced subgraphs. Our main result is a characterization of those complements of line graphs that are clique-perfect, also by means of minimal forbidden induced subgraphs. This implies an O(n²) time algorithm for deciding the clique-perfectness of complements of line graphs and, for those that are clique-perfect, finding α_c and τ_c.     

On natural isomorphisms of cycle permutation graphs

Two problems are approached in this paper. Given a permutation onn elements, which permutations onn elements yield cycle permutation graphs isomorphic to the cycle permutation graph yielded by the given permutation? And, given two cycle permutation graphs, are they isomorphic? Here the author deals only with natural isomorphisms, those isomorphisms which map the outer and inner cycles of one cycle permutation graph to the outer and inner cycles of another cycle permutation graph. A theorem is stated which then allows the construction of the set of permutations which yield cycle permutation graphs isomorphic to a given cycle permutation graph by a natural isomorphism. Another theorem is presented which finds the number of such permutations through the use of groups of symmetry of certain drawings of cycles in the plane. These drawings are also used to determine whether two given cycle permutation graphs are isomorphic by a natural isomorphism. These two methods are then illustrated by using them to solve the first problem, restricted to natural isomorphism, for a certain class of cycle permutation graphs.     

Independent domination and matchings in graphs

If G is a graph of order n, independent domination number i and matching number α₀, then i+α₀⩽n. We characterize all graphs for which equality holds in this inequality and show that this class can be recognized in polynomial time.     

Connected permutation graphs

A permutation graph is a simple graph associated with a permutation. Let c_n be the number of connected permutation graphs on n vertices. Then the sequence {c_n} satisfies an interesting recurrence relation such that it provides partitions of n! as . We also see that, if uniformly chosen at random, asymptotically almost all permutation graphs are connected.     

Cycle extendability in graphs and digraphs

In 1990, Hendry conjectured that all chordal Hamiltonian graphs are cycle extendable, that is, the vertices of each non-Hamiltonian cycle are contained in a cycle of length one greater. Let A be a symmetric (0,1)-matrix with zero main diagonal such that A is the adjacency matrix of a chordal Hamiltonian graph. Hendry’s conjecture in this case is that every k×k principle submatrix of A that dominates a full cycle permutation k×k matrix is a principle submatrix of a (k+1)×(k+1) principle submatrix of A that dominates a (k+1)×(k+1) full cycle permutation matrix. This article generalizes the concept of cycle-extendability to S-extendable; that is, with S⊆{1,2,…,n} and G a graph on n vertices, G is S-extendable if the vertices of every non-Hamiltonian cycle are contained in a cycle length i greater, where i∈S. We investigate this concept in directed graphs and in particular tournaments, i.e., anti-symmetric matrices with zero main diagonal.     

Comparability graphs and intersection graphs

A function diagram (f-diagram) D consists of the family of curves {$\text{1}\text{?}$ñ} obtained from n continuous functions $\text{f}{}_{\mathrm{i}}\text{:[0,1]→}\text{R}\text{(1?i?n)}$. We call the intersection graph of D a function graph (f-graph). It is shown that a graph G is an f-graph if and only if its complement ? is a comparability graph. An f-diagram generalizes the notion of a permulation diagram where the f_i are linear functions. It is also shown that G is the intersection graph of the concatenation of ?k permutation diagrams if and only if the partial order dimension of $\text{G}\text{?}\text{is ?k+1}$. Computational complexity results are obtained for recognizing such graphs.     

On n-skein isomorphisms of graphs

H. Whitney [Amer. J. Math.54 (1932), 150–168] proved that edge isomorphisms between connected graphs with at least five vertices are induced by isomorphisms and that circuit isomorphisms between 3-connected graphs are induced by isomorphisms. R. Halin and H. A. Jung [J. London Math. Soc.42 (1967), 254–256] generalized these results by showing that for n ≥ 2, n-skein isomorphisms between (n+1)-connected graphs are induced by isomorphisms. In this paper we show that for n ≥ 2, n-skein isomorphisms between 3-connected graphs having (n+1)-skeins are induced by isomorphisms.     

Two characterizations of interchange graphs of complete m-partite graphs

A graph G is m-partite if its points can be partitioned into m subsets V₁,…,V_m such that every line joins a point in V_i with a point in V_j, i ≠ j. A complete m-partite graph contains every line joining V_i with V_j. A complete graph K_p has every pair of its p points adjacent. The nth interchange graph I_n(G) of G is a graph whose points can be identified with the K_n+1's of G such that two points are adjacent whenever the corresponding K_n+1's have a K_n in common.Interchange graphs of complete 2-partite and 3-partite graphs have been characterized, but interchange graphs of complete m-partite graphs for m > 3 do not seem to have been investigated. The main result of this paper is two characterizations of interchange graphs of complete m-partite graphs for m ≥ 2.     

Well-covered graphs and factors

A maximum independent set of vertices in a graph is a set of pairwise nonadjacent vertices of largest cardinality α. Plummer [Some covering concepts in graphs, J. Combin. Theory 8 (1970) 91-98] defined a graph to be well-covered, if every independent set is contained in a maximum independent set of G. Every well-covered graph G without isolated vertices has a perfect [1,2]-factor F_G, i.e. a spanning subgraph such that each component is 1-regular or 2-regular. Here, we characterize all well-covered graphs G satisfying α(G)=α(F_G) for some perfect [1,2]-factor F_G. This class contains all well-covered graphs G without isolated vertices of order n with α?(n-1)/2, and in particular all very well-covered graphs.     

Rose window graphs underlying rotary maps

Given natural numbers n≥3 and 1≤a,r≤n−1, the rose window graph R_n(a,r) is a quartic graph with vertex set {x_i∣i∈Z_n}∪{y_i∣i∈Z_n} and edge set {{x_i,x_i+1}∣i∈Z_n}∪{{y_i,y_i+r}∣i∈Z_n}∪{{x_i,y_i}∣i∈Z_n}∪{{x_i+a,y_i}∣i∈Z_n}. In this paper rotary maps on rose window graphs are considered. In particular, we answer the question posed in [S. Wilson, Rose window graphs, Ars Math. Contemp. 1 (2008), 7-19. http://amc.imfm.si/index.php/amc/issue/view/5] concerning which of these graphs underlie a rotary map.     

Domination in permutation graphs

O(n³) algorithms to solve the weighted domination and weighted independent domination problems in permutation graphs, and an O(n²) algorithm to solve the cardinality domination problem in permutation graphs are presented.     

Long cycles and paths in distance graphs

For n∈N and D⊆N, the distance graph has vertex set {0,1,…,n−1} and edge set {ij∣0≤i,j≤n−1,|j−i|∈D}. Note that the important and very well-studied circulant graphs coincide with the regular distance graphs.A fundamental result concerning circulant graphs is that for these graphs, a simple greatest common divisor condition, their connectivity, and the existence of a Hamiltonian cycle are all equivalent. Our main result suitably extends this equivalence to distance graphs. We prove that for a finite set D of order at least 2, there is a constant c_D such that the greatest common divisor of the integers in D is 1 if and only if for every n, has a component of order at least n−c_D if and only if for every n≥c_D+3, has a cycle of order at least n−c_D. Furthermore, we discuss some consequences and variants of this result.     

A generalization of Turán's theorem to directed graphs

We consider an extremal problem for directed graphs which is closely related to Turán's theorem giving the maximum number of edges in a graph on n vertices which does not contain a complete subgraph on m vertices. For an integer n?2, let T_n denote the transitive tournament with vertex set X_n={1,2,3,…,n} and edge set {(i,j):1?i<j?n}. A subgraph H of T_n is said to be m-locally unipathic when the restriction of H to each m element subset of X_n consisting of m consecutive integers is unipathic. We show that the maximum number of edges in a m-locally unipathic subgraph of T_n is ($\text{q}\text{2}\text{)(m?1)}{}^2\text{+q(m?1)r+?}\text{1}\text{4}\text{r}{}^2\text{?}$ where n= q(m?1+r and $\text{?}\text{1}\text{2}\text{(m?1)??r<?}\text{3}\text{2}\text{(m?1)?}$. As is the case with Turán's theorem, the extremal graphs for our problem are complete multipartite graphs. Unlike Turán's theorem, the part sizes will not be uniform. The proof of our principal theorem rests on a combinatorial theory originally developed to investigate the rank of partially ordered sets.     

A combinatorial problem involving graphs and matrices

In this paper we discuss a combinatorial problem involving graphs and matrices. Our problem is a matrix analogue of the classical problem of finding a system of distinct representatives (transversal) of a family of sets and relates closely to an extremal problem involving 1-factors and a long standing conjecture in the dimension theory of partially ordered sets. For an integer n ?1, let n denote the n element set {1,2,3,…, n}. Then let A be a k×t matrix. We say that A satisfies property P(n, k) when the following condition is satisfied: For every k-taple (x₁,x₂,…,x_k?n^k there exist k distinct integers j₁,j₂,…,j_k so that x_i= a_ii for i= 1,2,…,k. The minimum value of t for which there exists a k × t matrix A satisfying property P(n,k) is denoted by f(n,k). For each k?1 and n sufficiently large, we give an explicit formula for f(n, k): for each n?1 and k sufficiently large, we use probabilistic methods to provide inequalities for f(n,k).     

Chromatic equivalence classes of complete tripartite graphs

Some necessary conditions on a graph which has the same chromatic polynomial as the complete tripartite graph K_m,n,r are developed. Using these, we obtain the chromatic equivalence classes for K_m,n,n (where 1≤m≤n) and K_m1,m2,m3 (where |m_i−m_j|≤3). In particular, it is shown that (i) K_m,n,n (where 2≤m≤n) and (ii) K_m1,m2,m3 (where |m_i−m_j|≤3, 2≤m_i,i=1,2,3) are uniquely determined by their chromatic polynomials. The result (i), proved earlier by Liu et al. [R.Y. Liu, H.X. Zhao, C.Y. Ye, A complete solution to a conjecture on chromatic uniqueness of complete tripartite graphs, Discrete Math. 289 (2004) 175-179], answers a conjecture (raised in [G.L. Chia, B.H. Goh, K.M. Koh, The chromaticity of some families of complete tripartite graphs (In Honour of Prof. Roberto W. Frucht), Sci. Ser. A (1988) 27-37 (special issue)]) in the affirmative, while result (ii) extends a result of Zou [H.W. Zou, On the chromatic uniqueness of complete tripartite graphs K_n1,n2,n3 J. Systems Sci. Math. Sci. 20 (2000) 181-186].     

Minimum number of edges in graphs that are both P2- and Pi-connected

A simple graph with n vertices is called P_i-connected if any two distinct vertices are connected by an elementary path of length i. In this paper, lower bounds of the number of edges in graphs that are both P₂- and P_i-connected are obtained. Namely if $\text{i?}\text{1}\text{2}\text{(n+1)}$, then |E(G)|?((4i?5)/(2i?2))(n?1), and if $\text{i >}\text{1}\text{2}\text{(n+ 1)}$, then |E(G)|?2(n?1) apart from one exeptional graph. Furthermore, extremal graphs are determined in the former.     

Maximum non-path-connected graphs

Let n and i be integers with n ≥ 4 and 2 ≤ i ≤ n ? 1. Lewin (J. Combin. Theory Ser. B25 (1978), 245–257) has determined the smallest size of a connected n-graph without end vertices which ensures the existence of a path of length i between every pair of distinct vertices of the graph. Here all the connected n-graps without end vertices of maximum size are found which fail to have a path of length i between every pair of distinct vertices.     

On short cycles in triangle-free oriented graphs

An orientation of a simple graph is referred to as an oriented graph. Caccetta and Häggkvist conjectured that any digraph on n vertices with minimum outdegree d contains a directed cycle of length at most ?n/d?. In this paper, we consider short cycles in oriented graphs without directed triangles. Suppose that α0 is the smallest real such that every n-vertex digraph with minimum outdegree at least α₀n contains a directed triangle. Let ε < (3 ? 2α₀)/(4 ? 2α₀) be a positive real. We show that if D is an oriented graph without directed triangles and has minimum outdegree and minimum indegree at least (1/(4 ? 2α₀)+ε)|D|, then each vertex of D is contained in a directed cycle of length l for each 4 ≤ l < (4 ? 2α₀)ε|D|/(3 ? 2α₀) + 2.