Marién Abreu

• University of Basilicata

A graph is pseudo 2-factor isomorphic if all of its 2-factors have the same parity of number of cycles. Abreu et al. [J. Comb. Theory, Ser. B. 98 (2008) 432--442] conjectured that $K_{3,3}$, the Heawood graph and the Pappus graph are the only essentially 4-edge-connected pseudo 2-factor isomorphic cubic bipartite graphs. This conjecture was disprov...

Let G be a graph of even order, and consider K G as the complete graph on the same vertex set as G. A perfect matching of K G is called a pairing of G. If for every pairing M of G it is possible to find a perfect matching N of G such that M ∪ N is a Hamiltonian cycle of K G , then G is said to have the Pairing-Hamiltonian property, or PH-property,...

A proper total colouring of a graph G is called harmonious if it has the further property that when replacing each unordered pair of incident vertices and edgeswith their colours, then no pair of colours appears twice. The smallest number of colours for it to exist is called the harmonious total chromatic number of G, denoted by ht(G). Here, we giv...

Let $G$ be a graph of even order, and consider $K_G$ as the complete graph on the same vertex set as $G$. A perfect matching of $K_G$ is called a pairing of $G$. If for every pairing $M$ of $G$ it is possible to find a perfect matching $N$ of $G$ such that $M \cup N$ is a Hamiltonian cycle of $K_G$, then $G$ is said to have the Pairing-Hamiltonian...

A graph $G$ has the Perfect-Matching-Hamiltonian property (PMH-property) if for each one of its perfect matchings, there is another perfect matching of $G$ such that the union of the two perfect matchings yields a Hamiltonian cycle of $G$. The study of graphs that have the PMH-property, initiated in the 1970s by Las Vergnas and Häggkvist, combines...

We study combinatorial configurations with the associated point and line graphs being strongly regular. Examples not belonging to known classes such as partial geometries and their generalizations or elliptic semiplanes are constructed. Necessary existence conditions are proved and a table of feasible parameters of such configurations with at most...

A graph $G$ admiting a $2$-factor is \textit{pseudo $2$-factor isomorphic} if the parity of the number of cycles in all its $2$-factors is the same. In [M. Abreu, A.A. Diwan, B. Jackson, D. Labbate and J. Sheehan. Pseudo $2$-factor isomorphic regular bipartite graphs. Journal of Combinatorial Theory, Series B, 98(2) (2008), 432-444.] some of the au...

A graph G has the Perfect-Matching-Hamiltonian property (PMH-property) if for each one of its perfect matchings, there is another perfect matching of G such that the union of the two perfect matchings yields a Hamiltonian cycle of G. The study of graphs that have the PMH-property, initiated in the 1970s by Las Vergnas and Häggkvist, combines three...

We study combinatorial configurations with the associated point and line graphs being strongly regular. Examples not belonging to the known families such as partial geometries and their generalizations or elliptic semiplanes are constructed. Necessary existence conditions are proved and a table of feasible parameters of such configurations with at...

The rook graph is a graph whose edges represent all the possible legal moves of the rook chess piece on a chessboard. The problem we consider is the following. Given any set $M$ containing pairs of cells such that each cell of the $m_{1} \times m_{2}$ chessboard is in exactly one pair, we determine the values of the positive integers $m_{1}$ and $m...

A graph admitting a perfect matching has the Perfect-Matching-Hamiltonian property (for short the PMH-property) if each of its perfect matchings can be extended to a Hamiltonian cycle. In this paper we establish some sufficient conditions for a graph G in order to guarantee that its line graph L(G) has the PMH-property. In particular, we prove that...

A graph admitting a perfect matching has the Perfect-Matching-Hamiltonian property (for short the PMH-property) if each of its perfect matchings can be extended to a Hamiltonian cycle. In this paper we establish some sufficient conditions for a graph G in order to guarantee that its line graph L(G) has the PMH-property. In particular, we prove that...

A graph has the Perfect-Matching-Hamiltonian property (for short the PMH-property) if each of its perfect matchings can be extended to a hamiltonian cycle. In this paper we establish some sufficient conditions for a graph $G$ in order to guarantee that its line graph $L(G)$ has the PMH-property. In particular, we prove that this happens when $G$ is...

A \emph{$k$--bisection} of a bridgeless cubic graph $G$ is a $2$--colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes have order at most $k$. Ban and Linial conjectured that {\em every bridgeless cubic graph admits a $2$--bisection except...

A \emph{$k$--bisection} of a bridgeless cubic graph $G$ is a $2$--colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes have order at most $k$. Ban and Linial conjectured that {\em every bridgeless cubic graph admits a $2$--bisection except...

A $k$-bisection of a bridgeless cubic graph $G$ is a $2$-colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes (monochromatic components in what follows) have order at most $k$. Ban and Linial conjectured that every bridgeless cubic graph ad...

A $k$-bisection of a bridgeless cubic graph $G$ is a $2$-colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes (monochromatic components in what follows) have order at most $k$. Ban and Linial conjectured that every bridgeless cubic graph ad...

We study snarks whose edges cannot be covered by fewer than five perfect matchings. Esperet and Mazzuoccolo found an infinite family of such snarks, generalising an example provided by Hagglund. We construct another infinite family, arising from a generalisation in a different direction. The proof that this family has the requested property is comp...

We give a characterization of Pfaffian graphs in terms of even orientations, extending the characterization of near bipartite non--pfaffian graphs by Fischer and Little \cite{FL}. Our graph theoretical characterization is equivalent to the one proved by Little in \cite{L73} (cf. \cite{LR}) using linear algebra arguments.

In this paper, we construct new infinite families of regular graphs of girth 7 of smallest order known so far. Our constructions are based on combinatorial and geometric properties of (q + 1, 8)-cages, for q a prime power. We remove vertices from such cages and add matchings among the vertices of minimum degree to achieve regularity in the new grap...

A graph G is 1-extendable if every edge belongs to at least one 1-factor. Let G be a graph with a 1-factor F. Then an even F-orientation of G is an orientation in which each F-alternating cycle has exactly an even number of edges directed in the same fixed direction around the cycle. In this paper, we examine the structure of 1-extendible graphs G...

Let $q\ge 2$ be a prime power. In this note we present a formulation for obtaining the known $(q+1,8)$-cages which has allowed us to construct small $(k,g)$--graphs for $k=q-1, q$ and $g=7,8$. Furthermore, we also obtain smaller $(q,8)$-graphs for even prime power $q$.

In this note we construct a new infinite family of $(q-1)$-regular graphs of girth $8$ and order $2q(q-1)^2$ for all prime powers $q\ge 16$, which are the smallest known so far whenever $q-1$ is not a prime power or a prime power plus one itself.

Viene descritto un esperimento di valorizzazione del patrimonio storico, archeologico e culturale realizzato attraverso un'originale integrazione di testo, immagini e musica. La conferenza-spettacolo Note Proteiche, presentata al Festival della Chimica di Potenza, segue un filo logico che unisce in un percorso ideale le seterie borboniche di San Le...

A snark is a cubic cyclically 4–edge connected graph with edge chromatic number four and girth at least five. We say that a graph G is odd 2–factored if for each 2–factor F of G each cycle of F is odd. In this extended abstract, we present a method for constructing odd 2–factored snarks. In particular, we construct two families of odd 2–factored sn...

Let 2≤r<m and g be positive integers. An ({r,m};g)-graph (or biregular graph) is a graph with degree set {r,m} and girth g, and an ({r,m};g)-cage (or biregular cage) is an ({r,m};g)-graph of minimum order n({r,m};g). If m=r+1, an ({r,m};g)-cage is said to be a semiregular cage. In this paper we generalize the reduction and graph amalgam operations...

The first known families of cages arised from the incidence graphs of generalized polygons of order $q$, $q$ a prime power. In particular, $(q+1,6)$--cages have been obtained from the projective planes of order $q$. Morever, infinite families of small regular graphs of girth 5 have been constructed performing algebraic operations on $\mathbb{F}_q$....

Let 2⩽r<m2⩽r<m and g be positive integers. An ({r,m};g)({r,m};g)–graph (or biregular graph) is a graph with degree set {r,m}{r,m} and girth g, and an ({r,m};g)({r,m};g)–cage (or biregular cage) is an ({r,m};g)({r,m};g)–graph of minimum order n({r,m};g)n({r,m};g). If m=r+1m=r+1, an ({r,m};g)({r,m};g)–cage is said to be a semiregular cage.In this ext...

A graph GG is pseudo 2-factor isomorphic if the parity of the number of cycles in a 2-factor is the same for all 2-factors of GG. In Abreu et al. (2008) [3] we proved that pseudo 2-factor isomorphic kk-regular bipartite graphs exist only for k≤3k≤3. In this paper we generalize this result for regular graphs which are not necessarily bipartite. We a...

A {\em snark} is a cubic cyclically 4-edge connected graph with edge chromatic number four and girth at least five. We say that a graph $G$ is {\em odd 2-factored} if for each 2-factor F of G each cycle of F is odd. In this paper, we present a method for constructing odd 2--factored snarks. In particular, we construct two families of odd 2-factored...

Let $k,l,m,n$, and $\mu$ be positive integers. A $\mathbb{Z}_\mu$--{\it scheme of valency} $(k,l)$ and {\it order} $(m,n)$ is a $m \times n$ array $(S_{ij})$ of subsets $S_{ij} \subseteq \mathbb{Z}_\mu$ such that for each row and column one has $\sum_{j=1}^n |S_{ij}| = k $ and $\sum_{i=1}^m |S_{ij}| = l$, respectively. Any such scheme is an algebra...

Let $q$ be a prime power; $(q+1,8)$-cages have been constructed as incidence graphs of a non-degenerate quadric surface in projective 4-space $P(4, q)$. The first contribution of this paper is a construction of these graphs in an alternative way by means of an explicit formula using graphical terminology. Furthermore by removing some specific perfe...

In this paper we give a method for obtaining the adjacency matrix of a simple polarity graph G q from a projective plane PG(2, q), where q is a prime power. Denote by ex(n; C 4) the maximum number of edges of a graph on n vertices and free of squares C 4. We use the constructed graphs G q to obtain lower bounds on the extremal function ex(n; C 4),...

A bipartite graph is pseudo 2-factor isomorphic if the number of circuits in each 2-factor of the graph is always even or always odd. We proved (Abreu etal., J Comb Theory B 98:432–442, 2008) that the only essentially 4-edge-connected pseudo 2-factor isomorphic cubic bipartite graph of girth 4 is K 3,3, and conjectured (Abreu etal., 2008, Conjectur...

In 1960, Hoffman and Singleton \cite{HS60} solved a celebrated equation for square matrices of order $n$, which can be written as $$ (\kappa - 1) I_n + J_n - A A^{\rm T} = A$$ where $I_n$, $J_n$, and $A$ are the identity matrix, the all one matrix, and a $(0,1)$--matrix with all row and column sums equal to $\kappa$, respectively. If $A$ is an inci...

We present three constructions which transform some symmetric config-uration K of type n k into new symmetric configurations of types (n + 1) k , or n k−1 , or ((λ − 1)µ) k−1 if n = λµ. Applying them to Desarguesian ellip-tic semiplanes, an infinite family of new configurations comes into being, whose types fill large gaps in the parameter spectrum...

We present three constructions which transform some symmetric configuration K of type n k into new symmetric configurations of types (n + 1) k , or n k−1 , or ((λ − 1)µ) k−1 if n = λµ. Applying them to Desarguesian ellip-tic semiplanes, an infinite family of new configurations comes into being, whose types fill large gaps in the parameter spectrum...

We point out several errors in our article [M. Abreu, R.E.L. Aldred, M. Funk, B. Jackson, D. Labbate, and J. Sheehan, ”Graphs and digraphs with all 2-factor isomorphic,” J. Comb. Theory, Ser. B 92, No. 2, 395–404 (2004; Zbl 1056.05109)] which were caused by our misquoting of a theorem of C. Thomassen. We also describe how the correct statement of T...

We present algebraic constructions yielding incidence matrices for all finite Desarguesian elliptic semiplanes of types C, D, and L. Both basic ingredients and suitable notations are derived from addition and multiplication tables of finite fields. This approach also applies to the only elliptic semiplane of type B known so far. In particular, the...

In this paper we investigate generalized circulant permutation matrices of composite order. We give a complete characterization of the order and the structure of symmetric generalized k-circulant permutation matrices in terms of circulant and retrocirculant block (0,1)-matrices in which each block contains exactly one or two entries 1. In particula...

Murty [A generalization of the Hoffman–Singleton graph, Ars Combin. 7 (1979) 191–193.] constructed a family of (pm+2)-regular graphs of girth five and order 2p2m, where p⩾5 is a prime, which includes the Hoffman–Singleton graph [A.J. Hoffman, R.R. Singleton, On Moore graphs with diameters 2 and 3, IBM J. (1960) 497–504]. This construction gives an...

A graph G is pseudo 2-factor isomorphic if the parity of the number of circuits in a 2-factor is the same for all 2-factors of G. We prove that there exist no pseudo 2-factor isomorphic k-regular bipartite graphs for k⩾4. We also propose a characterization for 3-edge-connected pseudo 2-factor isomorphic cubic bipartite graphs and obtain some partia...

Configurations of type (κ 2 +1) κ give rise to κ-regular simple graphs via configuration graphs. On the other hand, neighbourhood geometries of C 4 free κ-regular simple graphs on κ 2 +1 vertices turn out to be configurations of type (κ 2 +1) κ . We investigate which configurations of type (κ 2 +1) κ are equal or isomorphic to the neighbourhood geo...

We consider simple connected graphs for which there is a path of length at least?between every pair of distinct vertices. We wish to show that in these graphs the cycle space over ?2is generated by the cycles of length at leastmk, wherem= 1 for 3 ??? 6,m= 6/7 for?= 7,m? 1/2 for?? 8 andm? 3/4 +o(1) for large k.

We present three constructions which transform some symmetric configuration K of type n k into new symmetric configurations of types (n + 1) k , or n k−1 , or ((λ − 1)µ) k−1 if n = λµ. Applying them to Desarguesian ellip-tic semiplanes, an infinite family of new configurations comes into being, whose types fill large gaps in the parameter spectrum...

We consider simple connected graphs for which there is a path of length at least 6 between every pair of distinct vertices. We wish to show that in these graphs the cycle space over Z2 is generated by the cycles of length at least 6. Furthermore, we wish to generalize the result for k-path-connected graphs which contain a long cycle.

We show that a digraph which contains a directed 2-factor and has minimum in-degree and out-degree at least four has two non-isomorphic directed 2-factors. As a corollary, we deduce that every graph which contains a 2-factor and has minimum degree at least eight has two non-isomorphic 2-factors. In addition we construct: an infinite family of 3-dir...

A graph G is pseudo 2-factor isomorphic if the parity of the number of circuits in a 2-factor is the same for all 2-factors of G. In (1) we proved some results for pseudo 2-factor isomorphic regular bipartite graphs. In this paper we generalize some of those results for regular graphs which are not necessarily bipartite. We also introduce strongly...

We obtain some infinite families of graphs appearing as Cayley graphs in ℤ n and D n . Furthermore, a list of all the Cayley graphs coming from the groups up to order twelve is presented.

Typescript (Photocopy). Thesis (Ph. D.)--Florida Atlantic University, 2003. Includes bibliographical references (leaves 60-61).