Gabriela. Araujo-Pardo

• Dr
• Researcher at National Autonomous University of Mexico

A k-regular graph of girth g is called an edge-girth-regular graph, or an egr-graph for short, if each of its edges is contained in exactly λdistinct g-cycles. An egr-graph is called extremal for the triple (k, g, λ) if has the smallest possible order. We prove that some graphs arising from incidence graphs of finite planes are extremal egr-graphs....

For integers r ≥ 2, g ≥ 3 and χ ≥ 2, an (r, g, χ)-graph is an r-regular graph with girth g and chromatic number χ. Such a graph of minimum order is called an (r, g, χ)-cage. Here we prove the existence of (r, g, χ)-graphs for all r and even g when χ = 2 and for all r and g when χ = 3. Furthermore, using both existence proofs and explicit constructi...

In this paper we introduce a problem closely related to the Cage Problem and the Degree Diameter Problem. For integers k ≥ 2, g ≥ 3 and d ≥ 1, we define a (k; g, d)-graph to be a k-regular graph with girth g and diameter d. We denote by n₀(k; g, d) the smallest possible order of such a graph, and, if such a graph exists, we call it a (k; g, d)-cage...

In this paper, we obtain new lower and upper bounds for the problem of bipartite biregular cages. Moreover, for girth 6, we give the exact parameters of the (m, n; 6)-bipartite biregular cages when \(n\equiv -1\;\pmod m\) using the existence of a Steiner system \(S(2,k=m,v=1+n(m-1)+m)\). For girth \(g=2r\) and \(r=\{4,6,8\}\), we use results on t-g...

Cages ($r$-regular graphs of girth $g$ and minimum order) and their variants have been studied for over seventy years. Here we propose a new variant, "weighted cages". We characterize their existence; for cases $g=3,4$ we determine their order; we give Moore-like bounds and present some computational results.

A set of n non-collinear points in the Euclidean plane defines at least n different lines. Chen and Chv\'tal in 2008 conjectured that the same results is true in metric spaces for an adequate definition of line. More recently, it was conjectured in 2018 by Aboulker et al. that any large enough bridgeless graph on n vertices defines a metric space t...

The harmonious chromatic number of a graph G is the minimum number of colors that can be assigned to the vertices of G in a proper way such that any two distinct edges have different color pairs. This paper gives various results on harmonious chromatic number related to homomorphisms, incidence graphs of finite linear systems, and some circulant gr...

In this paper we study the achromatic arboricity of the complete graph. This parameter arises from the arboricity of a graph as the achromatic index arises from the chromatic index. The achromatic arboricity of a graph G, denoted by \(A_{\alpha }(G)\), is the maximum number of colors that can be used to color the edges of G such that every color cl...

An \emph{$(3,m;g)$ semicubic graph} is a graph in which all vertices have degrees either $3$ or $m$ and fixed girth $g$. In this paper, we construct families of semicubic graphs of even girth and small order using two different techniques. The first technique generalizes a previous construction which glues cubic cages of girth $g$ together at remot...

In this paper, we work with simple and finite graphs. We study a generalization of the \emph{Cage Problem}, which has been widely studied since cages were introduced by Tutte \cite{T47} in 1947 and after Erd\" os and Sachs \cite{ES63} proved their existence in 1963. An \emph{$(r,g)$-graph} is an $r$-regular graph in which the shortest cycle has len...

An edge-girth-regular graph egr(v,k,g,λ), is a k-regular graph of order v, girth g and with the property that each of its edges is contained in exactly λ distinct g-cycles. An egr(v,k,g,λ) is called extremal for the triple (k,g,λ) if v is the smallest order of any egr(v,k,g,λ). In this paper, we introduce two families of edge-girth-regular graphs....

A set of n non-collinear points in the Euclidean plane defines at least n different lines. Chen and Chv\'atal in 2008 conjectured that the same results is true in metric spaces for an adequate definition of line. More recently, this conjecture was studied in the context of quasi-metric spaces. In this work we prove that there is a quasi-metric spac...

The modelling of interconnection networks by graphs motivated the study of several extremal problems that involve well known parameters of a graph (degree, diameter, girth and order) and optimising one of the parameters given restrictions on some of the others. Here we focus on bipartite Moore graphs, that is, bipartite graphs attaining the optimum...

The harmonious chromatic number of a graph $G$ is the minimum number of colors that can be assigned to the vertices of $G$ in a proper way such that any two distinct edges have different color pairs. In this paper, we give various results on harmonious chromatic numbers related to homomorphisms, the incidence graph of finite linear systems, and the...

We extend the Grundy number and the ochromatic number, parameters on graph colorings, to digraph colorings, we call them {\emph{digrundy number}} and {\emph{diochromatic number}}, respectively. First, we prove that for every digraph the diochromatic number equals the digrundy number (as happens for graphs). Then, we prove the interpolation property...

A [z,r;g]-mixed cage is a mixed graph z-regular by arcs, r-regular by edges, with girth g and minimum order. Let n[z,r;g] denote the order of a [z,r;g]-mixed cage. In this paper we prove that n[z,r;g] is a monotonicity function, with respect to g, for z∈{1,2}, and we use it to prove that the underlying graph of a [z,r;g]-mixed cage is 2-connected,...

A bipartite biregular (m,n;g) $(m,n;g)$‐graph Γ ${\rm{\Gamma }}$ is a bipartite graph of even girth g $g$ having the degree set {m,n} $\{m,n\}$ and satisfying the additional property that the vertices in the same partite set have the same degree. An (m,n;g) $(m,n;g)$‐bipartite biregular cage is a bipartite biregular (m,n;g) $(m,n;g)$‐graph of minim...

A bipartite graph G=(V,E) with V=V1∪V2 is biregular if all the vertices of a stable set Vi have the same degree ri for i=1,2. In this paper, we give an improved new Moore bound for an infinite family of such graphs with odd diameter. This problem was introduced in 1983 by Yebra, Fiol, and Fàbrega. Besides, we propose some constructions of bipartite...

In a directed graph D, given two distinct vertices u and v, the line defined by the ordered pair (u,v) is the set of all vertices w such that u,v and w belong to a shortest directed path in D, containing a shortest directed path from u to v. In this work we study the following conjecture: the number of distinct lines in any strongly connected graph...

An edge-girth-regular graph $egr(v,k,g,\lambda)$, is a $k$-regular graph of order $v$, girth $g$ and with the property that each of its edges is contained in exactly $\lambda$ distinct $g$-cycles. An $egr(v,k,g,\lambda)$ is called extremal for the triple $(k,g,\lambda)$ if $v$ is the smallest order of any $egr(v,k,g,\lambda)$. In this paper, we int...

We consider the problem of finding long cycles in balanced tripartite graphs. We survey the relevant literature, namely degree and edge conditions for Hamiltonicity and long cycles in graphs, including bipartite and k-partite results where they exist. We then prove that if G is a balanced tripartite graph on 3n vertices, G must contain a cycle of l...

In this paper, we prove lower and upper bounds on the achromatic and the pseudoachromatic indices of the n-dimensional finite projective space of order q.

We extend the Grundy number and the ochromatic number, parameters on graph colorings, to digraph colorings, we call them {\emph{digrundy number}} and {\emph{diochromatic number}}, respectively. First, we prove that for every digraph the diochromatic number equals the digrundy number (as it happen for graphs). Then, we prove the interpolation proper...

In this paper we study the {\it {achromatic arboricity}} of the complete graph. This parameter arises from the arboricity of a graph as the achromatic index arises from the chromatic index. The achromatic arboricity of a graph $G$, denoted by $A_{\alpha}(G)$, is the maximum number of colors that can be used to color the edges of $G$ such that every...

A bipartite graph $G=(V,E)$ with $V=V_1\cup V_2$ is biregular if all the vertices of a stable set $V_i$ have the same degree $r_i$ for $i=1,2$. In this paper, we give an improved new Moore bound for an infinite family of such graphs with odd diameter. This problem was introduced in 1983 by Yebra, Fiol, and F\`abrega.\\ Besides, we propose some cons...

In this paper, we determine the achromatic and diachromatic numbers of some circulant graphs and digraphs each one with two lengths and give bounds for other circulant graphs and digraphs with two lengths. In particular, for the achromatic number we state that α (C16q2+20q+7(1, 2)) = 8q + 5, and for the diachromatic number we state that dac(C→\vec...

A \emph{$[z, r; g]$-mixed cage} is a mixed graph $z$-regular by arcs, $r$-regular by edges, with girth $g$ and minimum order. %In this paper we study structural properties of mixed cages: Let $n[z,r;g]$ denote the order of a $[z,r;g]$-mixed cage. In this paper we prove that $n[z,r;g]$ is a monotonicity function, with respect of $g$, for $z\in \{1,2...

A linear system is a pair $(P,\mathcal{L})$ where $\mathcal{L}$ is a family of subsets on a ground finite set $P$, such that $|l\cap l^\prime|\leq 1$, for every $l,l^\prime \in \mathcal{L}$. The elements of $P$ and $\mathcal{L}$ are called points and lines, respectively, and the linear system is called intersecting if any pair of lines intersect in...

The modelling of interconnection networks by graphs motivated the study of several extremal problems that involve well known parameters of a graph (degree, diameter, girth and order) and ask for the optimal value of one of them while holding the other two fixed. Here we focus in {\em bipartite Moore graphs\/}, that is, bipartite graphs attaining th...

Complete colorings have the property that any two color classes has at least an edge between them. Parameters such as the Grundy, achromatic and pseudoachromatic numbers comes from complete colorings, with some additional requirement. In this paper, we estimate these numbers in the Kneser graph $K(n,k)$ for some values of $n$ and $k$. We give the e...

In this paper, we determine the achromatic and diachromatic numbers of some circulant graphs and digraphs each one with two lengths and give bounds for other circulant graphs and digraphs with two lengths. In particular, for the achromatic number we state that $\alpha(C_{16q^2+20q+7}(1,2))=8q+5$, and for the diachromatic number we state that $dac(\...

In this work we present a version of the so called Chen and Chv\'atal's conjecture for directed graphs. A line of a directed graph D is defined by an ordered pair (u, v), with u and v two distinct vertices of D, as the set of all vertices w such that u, v, w belong to a shortest directed path in D containing a shortest directed path from u to v. A...

Let $G$ be a cubic graph and $\Pi$ be a polyhedral embedding of this graph. The extended graph, $G^{e},$ of $\Pi$ is the graph whose set of vertices is $V(G^{e})=V(G)$ and whose set of edges $E(G^{e})$ is equal to $E(G) \cup \mathcal{S}$, where $\mathcal{S}$ is constructed as follows: given two vertices $t_0$ and $t_3$ in $V(G^{e})$ we say $[t_0 t_...

We introduce the notion of a $[z, r; g]$-mixed cage. A $[z, r; g]$-mixed cage is a mixed graph $G$, $z$-regular by arcs, $r$-regular by edges, with girth $g$ and minimum order. In this paper we prove the existence of $[z, r ;g]$-mixed cages and exhibit families of mixed cages for some specific values. We also give lower and upper bounds for some ch...

A bipartite biregular $(n,m;g)$-graph $G$ is a bipartite graph of even girth $g$ having the degree set $\{n,m\}$ and satisfying the additional property that the vertices in the same partite set have the same degree. An $(n,m;g)$-bipartite biregular cage is a bipartite biregular $(n,m;g)$-graph of minimum order. In their 2019 paper, Filipovski, Ramo...

A decomposition of a simple graph G is a pair (G, P) where P is a set of subgraphs of G, which partitions the edges of G in the sense that every edge of G belongs to exactly one subgraph in P. If the elements of P are induced subgraphs then the decomposition is denoted by [G, P ]. A k-P-coloring of a decomposition (G, P) is a surjective function th...

A linear system is a pair $(P,\mathcal{L})$ where $\mathcal{L}$ is a family of subsets on a ground finite set $P$, such that $|l\cap l^\prime|\leq 1$, for every $l,l^\prime \in \mathcal{L}$. The elements of $P$ and $\mathcal{L}$ are called points and lines, respectively, and the linear system is called intersecting if any pair of lines intersect in...

In this paper, we study the achromatic and the pseudoachromatic numbers of planar and outerplanar graphs as well as planar graphs of girth 4 and graphs embedded on a surface. We give asymptotically tight results and lower bounds for maximal embedded graphs.

A line-coloring of the finite affine space $\mathrm{AG}(n,q)$ is \emph{proper} if any two lines from the same color class have no point in common, and it is \emph{complete} if for any two different colors $i$ and $j$ there exist two intersecting lines, one is colored by $i$ and the other is colored by $j.$ The pseudoachromatic index of $\mathrm{AG...

We study the Cage Problem for regular and biregular planar graphs. A $(k,g)$-graph is a $k$-regular graph with girth $g$. A $(k,g)$-cage is a $(k,g)$-graph of minimum order. It is not difficult to conclude that the regular planar cages are the Platonic Solids. A $(\{r,m\};g)$-graph is a graph of girth $g$ whose vertices have degrees $r$ and $m.$ A...

In this paper, we generalize the concept of \emph{perfect graphs} to other parameters related to graph vertex coloring. This idea was introduced by Christen and Selkow in 1979 and Yegnanarayanan in 2001. Let $ a,b \in \{ \omega, \chi, \Gamma, \alpha, \psi \} $ where $ \omega $ is the clique number, $ \chi $ is the chromatic number, $ \Gamma $ is th...

We consider the extension to directed graphs of the concept of the achromatic number in terms of acyclic vertex colorings. The achromatic number has been intensely studied since it was introduced by Harary, Hedetniemi and Prins in 1967. The dichromatic number is a generalization of the chromatic number for digraphs defined by Neumann-Lara in 1982....

We consider the extension to directed graphs of the concept of the achromatic number in terms of acyclic vertex colorings. The achromatic number has been intensely studied since it was introduced by Harary, Hedetniemi and Prins in 1967. The dichromatic number is a generalization of the chromatic number for digraphs defined by Neumann-Lara in 1982....

We consider the extension to directed graphs of the concept of the achromatic number in terms of acyclic vertex colorings. The achromatic number has been intensely studied since it was introduced by Harary, Hedetniemi and Prins in 1967. The dichromatic number is a generalization of the chromatic number for digraphs defined by Neumann-Lara in 1982....

Let Π_q be the projective plane of order q, let Ψ(m):=Ψ(L(K_m)) the pseudoachromatic number of the complete line graph of order m, let a∈{3,4,...,q/2+1} and m_a=(q+1)²-a. In this paper, we improve the upper bound of Ψ(m) given by Araujo-Pardo et al. [J Graph Theory 66 (2011), 89-97] and Jamison [Discrete Math. 74 (1989), 99-115] in the following va...

The pseudoachromatic index of the finite affine space $\mathrm{AG}(n,q),$ denoted by $\psi'(\mathrm{AG}(n,q)),$ is the the maximum number of colors in any complete line-coloring of $\mathrm{AG}(n,q).$ When the coloring is also proper, the maximum number of colors is called the achromatic index of $\mathrm{AG}(n,q).$ We prove that if $n$ is even the...

A 'complete $k$-coloring' of a graph $G$ is a (not necessarily proper) $k$-coloring of the vertices of $G$ such that each pair of different colors appears in an edge. A complete $k$-coloring is also called 'connected', if each color class induces a connected subgraph of $G$. The 'pseudoachromatic index' of a graph $G$, denoted by $\psi'(G)$, is the...

A linear system is a pair $(X,\mathcal{F})$ where $\mathcal{F}$ is a finite family of subsets on a ground set $X$, and it satisfies that $|A\cap B|\leq 1$ for every pair of distinct subsets $A,B \in \mathcal{F}$. As an example of a linear system are the straight line systems, which family of subsets are straight line segments on $\mathbb{R}^{2}$. B...

A (k,g)-graph is a k-regular graph with girth g and a (k,g)-cage is a (k,g)-graph with the fewest possible number of vertices. The cage problem consists of constructing (k,g)-graphs of minimum order n(k,g). We focus on girth g=5, where cages are known only for degrees k≤7. We construct (k,5)-graphs using techniques exposed by Funk (2009) and Abreu...

Let $D$ be a connected oriented graph. A set $S \subseteq V(D)$ is convex in $D$ if, for every pair of vertices $x, y \in S$, the vertex set of every $xy$-geodesic, ($xy$ shortest directed path) and every $yx$-geodesic in $D$ is contained in $S$. The convexity number, ${\rm con}(D)$, of a non-trivial oriented graph, $D$, is the maximum cardinality...

In this paper, we study the achromatic and the pseudoachromatic numbers of planar and outerplanar graphs as well as planar graphs of girth 4 and graphs embedded on a surface. We give asymptotically tight results and lower bounds for maximal embedded graphs.

Let $D$ be a connected oriented graph. A set $S \subseteq V(D)$ is convex in $D$ if, for every pair of vertices $x, y \in S$, the vertex set of every $xy$-geodesic, ($xy$ shortest directed path) and every $yx$-geodesic in $D$ is contained in $S$. The convexity number, ${\rm con}(D)$, of a non-trivial oriented graph, $D$, is the maximum cardinality...

We introduce the notion of a $[z, r; g]$-mixed cage. A $[z, r; g]$-mixed cage is a mixed graph $G$, $z$-regular by arcs, $r$-regular by edges, with girth $g$ and minimum order. In this paper we prove the existence of $[z, r ;g]$-mixed cages and exhibit families of mixed cages for some specific values. We also give lower and upper bounds for some ch...

In this paper, we prove lower and upper bounds on the achromatic and the pseudoachromatic indices of the $n$-dimensional finite projective space of order $q$.

A linear system is a pair (X, F) where F is a finite family of subsets on a ground set X, and it satisfies that |A N B| ≤ 1 for every pair of distinct subsets A, B ϵ T. As an example of a linear system are the straight line systems, which family of subsets are straight line segments on R². By r and 1/2 we denote the size of the minimal transversal...

A mixed regular graph is a connected simple graph in which each vertex has both a fixed outdegree (the same indegree) and a fixed undirected degree. A mixed regular graphs is said to be optimal if there is not a mixed regular graph with the same parameters and bigger order. We present a construction that provides mixed graphs of undirected degree q...

A -cage is a k-regular graph of girth g of minimum order. In this work, we focus on girth , where cages are known only for degrees . When , except perhaps for , the order of a -cage is strictly greater than . Considering the relationship between finite geometries and graphs we establish upper constructive bounds that improve the best so far.

A mixed graph is said to be dense, if its order is close to the Moore bound and it is optimal if there is not a mixed graph with the same parameters and bigger order. We give a construction that provides dense mixed graphs of undirected degree q, directed degree and order , for q being an odd prime power. Since the Moore bound for a mixed graph wit...

A linear hypergraph is intersecting if any two different edges have exactly one common vertex and an $n$-quasicluster is an intersecting linear hypergraph with $n$ edges each one containing at most $n$ vertices and every vertex is contained in at least two edges. The Erd\"os-Faber-Lov\'asz Conjecture states that the chromatic number of any $n$-quas...

The achromatic number α of a graph is the largest number of colors that can be assigned to its vertices such that adjacent vertices have different color and every pair of different colors appears on the end vertices of some edge. We estimate the achromatic number of Kneser graphs K(n, k) and determine α(K(n, k)) for some values of n and k. Furtherm...

The pseudoachromatic index of a graph is the maximum number of colors that can be assigned to its edges, such that each pair of different colors is incident to a common vertex. If for each vertex its incident edges have different color, then this maximum is known as achromatic index. Both indices have been widely studied. A geometric graph is a gra...

A graph of girth g that contains vertices of degrees r and m is called a bi-regular ({r,m},g)-graph. As with the Cage Problem, we seek the smallest ({r,m},g)-graphs for given parameters 2≤<m, g≥3, called ({r,m},g)-cages. The orders of the majority of ({r,m},g)-cages, in cases where m is much larger than r and the girth g is odd, have been recently...

A mixed graph is said to be dense if its order is close to the Moore bound and it is optimal if there is not a mixed graph with the same parameters and bigger order. We present a construction that provides dense mixed graphs of undirected degree $q$, directed degree $\frac{q-1}{2}$ and order $2q^2$, for $q$ being an odd prime power. Since the Moore...

A \emph{linear system} is a pair $(X,\mathcal{F})$ where $\mathcal{F}$ is a iinite family of subsets on a ground set $X$, and it satisfies that $|A\cap B|\leq 1$ for every pair of distinct subsets $A,B \in \mathcal{F}$. As an example of a linear system are the straight line systems, which family of subsets are straight line segments on $\mathbb{R}^...

A decomposition of a simple graph G is a pair (G,D) where D is a set of subgraphs of G, which partitions the edges of G in the sense that every edge of G belongs to exactly one subgraph in D. If the elements of D are induced subgraphs then the decomposition is denoted by [G,D]. A k-D-coloring of a decomposition (G,D) is a surjective function that a...

We study vertex colorings of hypergraphs, such that all color sizes differ at most in one (balanced colorings) and each edge contains at least two vertices of the same color (rainbow-free colorings). Given a hypergraph H, the maximum k, such that there is a balanced rainbow-free k-coloring of H is called the balanced upper chromatic number denoted...

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...

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.

Every n-edge colored n-regular graph G naturally gives rise to a simple abstract n-polytope, the colorful polytope of G, whose 1-skeleton is isomorphic to G. The paper describes colorful polytope versions of the associahedron and cyclohedron. Like their classical counterparts, the colorful associahedron and cyclohedron encode triangulations and fli...

The problem of computing the chromatic number of Kneser hypergraphs has been extensively studied over the last 40 years and the fractional version of the chromatic number of Kneser hypergraphs is only solved for particular cases. The \emph{$(p,q)$-extremal problem} consists in finding the maximum number of edges on a $k$-uniform hypergraph $\mathca...

Let $ \Pi_q $ be the projective plane of order $ q $, let $\psi(m):=\psi(L(K_m))$ the pseudoachromatic number of the complete line graph of order $ m $, let $ a\in \{ 3,4,\dots,\tfrac{q}{2}+1 \} $ and $ m_a=(q+1)^2-a $. In this paper, we improve the upper bound of $ \psi(m) $ given by Araujo-Pardo et al. [J Graph Theory 66 (2011), 89--97] and Jamis...

Let r, m, 2 ≤ r m and g ≥ 3 be three positive integers. A graph with a prescribed degree set r, m and girth g having the least possible number of vertices is called a bi-regular cage or an (r, m; g)-cage, and its order is denoted by n(r, m; g). In this paper we provide upper bounds on n(r, m; g) for some related values of r, m and even girth g at l...

In this paper we study a natural generalization for the perfection of graphs to other interesting parameters related with colorations. This generalization was introduced partially by Christen and Selkow in 1979 and Yegnanarayanan in 2001. Let a,b∈{ω,χ,Γ,α,ψ}a,b∈{ω,χ,Γ,α,ψ} where ω is the clique number, χ is the chromatic number, Γ is the Grundy num...

Let 2⩽r<m2⩽r<m and g⩾4g⩾4 even be three positive integers. A graph with a degree set {r,m}{r,m}, girth g and minimum order is called a bi-regular cage or an ({r,m};g)({r,m};g)-cage, and its order is denoted by n({r,m};g)n({r,m};g). In this paper we obtain constructive upper bounds on n({r,m};g)n({r,m};g) for some values of r,mr,m and even girth at...

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 pseudoachromatic index of a graph is the maximum number of colors that can be assigned to its edges, such that each pair of different colors is incident to a common vertex. If for each vertex its incident edges have different color, then this maximum is known as achromatic index. Both indices have been widely studied. A geometric graph is a gra...

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...

The paper investigates connections between abstract polytopes and properly edge colored graphs. Given any finite n-edge-colored n-regular graph G, we associate to G a simple abstract polytope P G of rank n, the colorful polytope of G, with 1-skeleton isomorphic to G. We investigate the interplay between the geometric, combinatorial, or algebraic pr...

Given a vertex v of a graph G the second order degree of v denoted as d 2(v) is defined as the number of vertices at distance 2 from v. In this paper we address the following question: What are the sufficient conditions for a graph to have a vertex v such that d 2(v) ≥ d(v), where d(v) denotes the degree of v? Among other results, every graph of mi...

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...

A hypergraph is linear if any two different edges intersect in at most one vertex. The Erdös-Faber-Lovász Conjecture may be stated as follow: Any linear hypergraph H consisting of n edges each one containing n vertices has chromatic number n . In the present poster we present the correctness of the conjecture for a new infinite class of linear hype...

Let (P,L,I) be a partial linear space and X⊆P∪L. Let us denote (X)I=⋃x∈X{y:yIx} and [X]=(X)I∪X. With this terminology a partial linear space(P,L,I)is said to admit a(1,≤k)-identifying code if and only if the sets [X] are mutually different for all X⊆P∪L with ∣X∣≤k. In this paper we give a characterization of k-regular partial linear spaces admittin...

Let q = 2β be, for some β∈ℕ, and let n = q2 + q+ 1. By exhibiting a complete coloring of the edges of Kn, we show that the pseudoachromatic number ψ(Gn) of the complete line graph Gn = L(Kn)—or the pseudoachromatic index of Kn, if you will—is at least q3 + q. This bound improves the implicit bound of Jamison [Discrete Math 74 (1989), 99–115] which...

Let (P , L, I) be a partial linear space and X ⊆ P ∪L. Let us denote (X) I =  x∈X {y : yIx} and [X] = (X) I ∪ X. With this terminology a partial linear space (P , L, I) is said to admit a (1, ≤k)-identifying code if and only if the sets [X] are mutually different for all X ⊆ P ∪ L with | X |≤ k. In this paper we give a characterization of k-regula...

In this article, some structures in the projective plane of order are found which allow us to construct small -regular balanced bipartite graphs of girth 6 for all . When , the order of these -regular graphs is ; and when , the order of these -regular graphs is . Moreover, the incidence matrix of a -regular balanced bipartite graph of girth 6 havin...

We give a construction of k-regular graphs of girth g using only geometrical and combinatorial properties that appear in any (k;g+1)-cage, a minimal k-regular graph of girth g+1. In this construction, g≥5 and k≥3 are odd integers, in particular when k−1 is a power of 2 and (g+1)∈{6,8,12} we use the structure of generalized polygons. With this const...

Small k-regular graphs of girth g where g=6,8,12 are obtained as subgraphs of minimal cages. More precisely, we obtain (k,6)-graphs on 2(kq−1) vertices, (k,8)-graphs on 2k(q2−1) vertices and (k,12)-graphs on 2kq2(q2−1), where q is a prime power and k is a positive integer such that q≥k≥3. Some of these graphs have the smallest number of vertices kn...

In this paper, we exhibit infinite families of vertex critical r-dichromatic circulant tournaments for all r >= 3. The existence of these infinite families was conjectured by Neumann-Lara [V. Neumann-Lara, Note on vertex critical 4-dichromatic circulant tournaments, Discrete Math. 170 (1997) 289-291], who later proved it for all r >= 3 and r not eq...