Coloring perfect (K4 − e)-free graphs

This note proves the Strong Perfect Graph Conjecture for (K₄ − e)-free graphs from first principles. The proof directly yields an O(pn²) algorithm for p-coloring a perfect (K₄ − e)-free graph.     

Characterizations for co-graphs defined by restricted NLC-width or clique-width operations

In this paper we characterize subclasses of co-graphs defined by restricted NLC-width operations and subclasses of co-graphs defined by restricted clique-width operations.We show that a graph has NLCT-width 1 if and only if it is (C₄,P₄)-free. Since (C₄,P₄)-free graphs are exactly trivially perfect graphs, the set of graphs of NLCT-width 1 is equal to the set of trivially perfect graphs, and a recursive definition for trivially perfect graphs follows. Further we show that a graph has linear NLC-width 1 if and only if is (C₄,P₄,2K₂)-free. This implies that the set of graphs of linear NLC-width 1 is equal to the set of threshold graphs.We also give forbidden induced subgraph characterizations for co-graphs defined by restricted clique-width operations using P₄, 2K₂, and co-2P₃.     

Perfect state transfer in integral circulant graphs of non-square-free order

This paper provides further results on the perfect state transfer in integral circulant graphs (ICG graphs). The non-existence of PST is proved for several classes of ICG graphs containing an isolated divisor d₀, i.e. the divisor which is relatively prime to all other divisors from d∈D?{d₀}. The same result is obtained for classes of integral circulant graphs having the NSF property (i.e. each n/d is square-free, for every d∈D). A direct corollary of these results is the characterization of ICG graphs with two divisors, which have PST. A similar characterization is obtained for ICG graphs where each two divisors are relatively prime. Finally, it is shown that ICG graphs with the number of vertices n=2p² do not have PST.     

On the non-existence of perfect and nearly perfect codes

The main result of the paper is the proof of the non-existence of a class of completely regular codes in certain distance-regular graphs. Corollaries of this result establish the non-existence of perfect and nearly perfect codes in the infinite families of distance-regular graphs J(2b + 1, b) and J(2b+2,b).     

On the theta number of powers of cycle graphs

We give a closed formula for Lovász’s theta number of the powers of cycle graphs C _k ^d?1 and of their complements, the circular complete graphs K _k/d . As a consequence, we establish that the circular chromatic number of a circular perfect graph is computable in polynomial time. We also derive an asymptotic estimate for the theta number of C _k ^d .     

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.     

Enumeration of perfect matchings of a type of Cartesian products of graphs

Let G be a graph and let Pm(G) denote the number of perfect matchings of G.We denote the path with m vertices by P_m and the Cartesian product of graphs G and H by G×H. In this paper, as the continuance of our paper [W. Yan, F. Zhang, Enumeration of perfect matchings of graphs with reflective symmetry by Pfaffians, Adv. Appl. Math. 32 (2004) 175-188], we enumerate perfect matchings in a type of Cartesian products of graphs by the Pfaffian method, which was discovered by Kasteleyn. Here are some of our results:1. Let T be a tree and let C_n denote the cycle with n vertices. Then Pm(C₄×T)=∏(2+α²), where the product ranges over all eigenvalues α of T. Moreover, we prove that Pm(C₄×T) is always a square or double a square.2. Let T be a tree. Then Pm(P₄×T)=∏(1+3α²+α⁴), where the product ranges over all non-negative eigenvalues α of T.3. Let T be a tree with a perfect matching. Then Pm(P₃×T)=∏(2+α²), where the product ranges over all positive eigenvalues α of T. Moreover, we prove that Pm(C₄×T)=[Pm(P₃×T)]².     

An application of the structural theory of acyclic skew-symmetric graphs

It is known that if an undirected graph G contains a unique perfect matching M, then M contains at least one of bridges of G. In this paper an alternative proof of this statement is presented. The proof is based on the structural theory of acyclic skew-symmetric graphs developed by the author.     

Neighborhood perfect graphs

The notion of neighborhood perfect graphs is introduced here as follows. Let G be a graph, α_N(G) denote the maximum number of edges such that no two of them belong to the same subgraph of G induced by the (closed) neighborhood of some vertex; let ϱ_N(G) be the minimum number of vertices whose neighborhood subgraph cover the edge set of G. Then G is called neighborhood perfect if ϱ_N(G′) = α_N(G′) holds for every induced subgraph G′ of G. It is expected that neighborhood perfect graphs are perfect also in the sense of Berge. We characterized here those chordal graphs which are neighborhood perfect. In addition, an algorithm to computer ϱ_N(G) = α_N(G) is given for interval graphs.     

Perfect Matchings of Cellular Graphs

We introduce a family of graphs, called cellular, and consider the problem of enumerating their perfect matchings. We prove that the number of perfect matchings of a cellular graph equals a power of 2 times the number of perfect matchings of a certain subgraph, called the core of the graph. This yields, as a special case, a new proof of the fact that the Aztec diamond graph of order n introduced by Elkies, Kuperberg, Larsen and Propp has exactly 2 ^n(n+1)/2 perfect matchings. As further applications, we prove a recurrence for the number of perfect matchings of certain cellular graphs indexed by partitions, and we enumerate the perfect matchings of two other families of graphs called Aztec rectangles and Aztec triangles.     

Ear-decompositions of matching-covered graphs

We call a graphmatching-covered if every line belongs to a perfect matching. We study the technique of “ear-decompositions” of such graphs. We prove that a non-bipartite matching-covered graph containsK ₄ orK ₂⊕K ₃ (the triangular prism). Using this result, we give new characterizations of those graphs whose matching and covering numbers are equal. We apply these results to the theory of τ-critical graphs.     

On forcing matching number of boron-nitrogen fullerene graphs

Let G be a graph that admits a perfect matching M. A forcing set S for a perfect matching M is a subset of M such that it is contained in no other perfect matchings of G. The smallest cardinality of forcing sets of M is called the forcing number of M. Computing the minimum forcing number of perfect matchings of a graph is an NP-complete problem. In this paper, we consider boron-nitrogen (BN) fullerene graphs, cubic 3-connected plane bipartite graphs with exactly six square faces and other hexagonal faces. We obtain the forcing spectrum of tubular BN-fullerene graphs with cyclic edge-connectivity 3. Then we show that all perfect matchings of any BN-fullerene graphs have the forcing number at least two. Furthermore, we mainly construct all seven BN-fullerene graphs with the minimum forcing number two.     

q-Coverings,codes, and line graphs

In this paper we consider the relationship between q-coverings of a regular graph and perfect 1-codes in line graphs. An infinite class of perfect 1-codes in the line graphs L(I_k) is constructed.     

On almost self-complementary graphs

A graph is called almost self-complementary if it is isomorphic to one of its almost complements X^c-I, where X^c denotes the complement of X and I a perfect matching (1-factor) in X^c. Almost self-complementary circulant graphs were first studied by Dobson and Šajna [Almost self-complementary circulant graphs, Discrete Math. 278 (2004) 23-44]. In this paper we investigate some of the properties and constructions of general almost self-complementary graphs. In particular, we give necessary and sufficient conditions on the order of an almost self-complementary regular graph, and construct infinite families of almost self-complementary regular graphs, almost self-complementary vertex-transitive graphs, and non-cyclically almost self-complementary circulant graphs.     

A note on Ki-perfect graphs

Ki-perfect graphs are a special instance of F - G perfect graphs, where F and G are fixed graphs with F a partial subgraph of G. Given S, a collection of G-subgraphs of graph K, an F - G cover of S is a set of T of F-subgraphs of K such that each subgraph in S contains as a subgraph a member of T. An F - G packing of S is a subcollection S′? S such that no two subgraphs in S′ have an F-subgraph in common. K is F - G perfect if for all such S, the minimum cardinality of an F - G cover of S equals the maximum cardinality of an F - G packing of S. Thus K_i-perfect graphs are precisely K_i-1 - K_i perfect graphs. We develop a hypergraph characterization of F - G perfect graphs that leads to an alternate proof of previous results on K_i-perfect graphs as well as to a characterization of F - G perfect graphs for other instances of F and G.     

Strongly regular graphs associated with ternary bent functions

We prove a new characterization of weakly regular ternary bent functions via partial difference sets. Partial difference sets are combinatorial objects corresponding to strongly regular graphs. Using known families of bent functions, we obtain in this way new families of strongly regular graphs, some of which were previously unknown. One of the families includes an example in [N. Hamada, T. Helleseth, A characterization of some {3v₂+v₃,3v₁+v₂,3,3}-minihypers and some [15,4,9;3]-codes with B₂=0, J. Statist. Plann. Inference 56 (1996) 129-146], which was considered to be sporadic; using our results, this strongly regular graph is now a member of an infinite family. Moreover, this paper contains a new proof that the Coulter-Matthews and ternary quadratic bent functions are weakly regular.     

Graphs with integral spectrum

It is shown that only a fraction of 2^-Ω(n) of the graphs on n vertices have an integral spectrum. Although there are several explicit constructions of such graphs, no upper bound for their number has been known. Graphs of this type play an important role in quantum networks supporting the so-called perfect state transfer.     

Sorting with networks of data structures

We consider the problem of sorting a permutation using a network of data structures as introduced by Knuth and Tarjan. In general the model as considered previously was restricted to networks that are directed acyclic graphs (DAGs) of stacks and/or queues. In this paper we study the question of which are the smallest general graphs that can sort an arbitrary permutation and what is their efficiency. We show that certain two-node graphs can sort in time Θ(n²) and no simpler graph can sort all permutations. We then show that certain three-node graphs sort in time Ω(n^3/2), and that there exist graphs of k nodes which can sort in time Θ(nlog_kn), which is optimal.     

Perfect codes in the graphs Ok

In this paper we consider the existence of perfect codes in the infinite class of distance-transitive graphs O_k. Perfect 1-codes correspond to certain Steiner systems and necessary conditions for the existence of such a code are satisfied if k + 1 is prime. We give some nonexistence results for perfect 2-, 3-, and 4-codes and for perfect e-codes in general, including a lower bound for k in terms of e.