Combinatorics (math.CO)

Abstract: In the present paper, we give proofs of the existence of a 3-design in the extended ternary quadratic residue code of length 14 and the extended quaternary quadratic residue code of length 18.

Abstract: 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 remote vertices (vertices at distance at least $g/2$). The second technique, the main content of this paper, produces bipartite semicubic $(3,m; g)$-graphs with fixed even girth $g = 4t$ or $4t+2$ using voltage graphs over $\mathbb{Z}_{m}$. When $g = 4t+2$, the graphs have two vertices of degree $m$, while when $g = 4t$ they have exactly three vertices of degree $m$ (the remaining vertices are of degree $3$ in both cases). Specifically, we describe infinite families of semicubic graphs $(3,m; g)$ for $g = \{6, 8, 10, 12\}$ for infinitely many values of $m$. The cases $g = \{6,8\}$ include the unique $6$-cage and the unique $8$-cage when $m = 3$. The families obtained in this paper for girth $g=\{10,12\}$ include examples with the best known bounds for semicubic graphs $(3,m; g)$

Abstract: Brown et al. provide a representation of a spread of the Tits quadrangle $T_2(\mathcal O)$, $\mathcal O$ an oval of $\mathrm PG(2,q)$, $q$ even, in terms of a certain family of $q$ ovals of $\mathrm PG(2,q)$. By combining this representation with the Vandendriessche classification of hyperovals in $\mathrm PG(2,64)$ and the classification of flocks of the quadratic cone in $\mathrm PG(3,64)$, recently given by the authors, in this paper, we classify all the spreads of $T_2(\mathcal O)$, $\mathcal O$ an oval of $\mathrm PG(2,64)$, up to equivalence. These complete the classification of spreads of $T_2(\mathcal O)$ for $q\le 64$.

Abstract: A $d$-dimensional framework is a pair $(G,p)$, where $G$ is a graph and $p$ maps the vertices of $G$ to points in $\mathbb{R}^d$. The edges of $G$ are mapped to the corresponding line segments. A graph $G$ is said to be globally rigid in $\mathbb{R}^d$ if every generic $d$-dimensional framework $(G,p)$ is determined, up to congruence, by its edge lengths. A finer property is global linkedness: we say that a vertex pair $\{u,v\}$ of $G$ is globally linked in $G$ in $\mathbb{R}^d$ if in every generic $d$-dimensional framework $(G,p)$ the distance of $u$ and $v$ is uniquely determined by the edge lengths. In this paper we investigate globally linked pairs in graphs in $\mathbb{R}^d$. We give several characterizations of those rigid graphs $G$ in which a pair $\{u,v\}$ is globally linked if and only if there exist $d+1$ internally disjoint paths from $u$ to $v$ in $G$. We call these graphs $d$-joined. Among others, we show that $G$ is $d$-joined if and only if for each pair of generic frameworks of $G$ with the same edge lengths, one can be obtained from the other by a sequence of partial reflections along hyperplanes determined by $d$-separators of $G$. We also show that the family of $d$-joined graphs is closed under edge addition, as well as under gluing along $d$ or more vertices. As a key ingredient to our main results, we prove that rigid graphs in $\mathbb{R}^d$ contain no crossing $d$-separators. Our results give rise to new families of graphs for which global linkedness (and global rigidity) in $\mathbb{R}^d$ can be tested in polynomial time.

Abstract: For a graph $G$ with the vertex set $V(G)$ and the edge set $E(G)$ and a star subgraph $S$ of $G$, let $\alpha_S(G)$ be the maximum number of vertices in $G$ such that no two of them are in the same star subgraph $S$ and $\theta_S(G)$ be the minimum number of star subgraph $S$ that cover the vertices of $G$. A graph $G$ is called $S$-perfect if for every induced subgraph $H$ of $G$, $\alpha_S(H)=\theta_S(H)$. Motivated by perfect graphs discovered by Berge, Ravindra introduced $S$-perfect graphs. In this paper we prove that a triangulated graph is $S$-perfect if and only if $G$ is odd sun-free. This result leads to a conjecture which if proved is a structural characterization of $S$-perfect graphs in terms of forbidden subgraphs.

Abstract: A partition $\pi=\{V_{1},V_{2},...,V_{k}\}$ of the vertex set $V$ of a graph $G$ into $k$ color classes $V_{i}$, with $1\leq i\leq k$ is called a quorum coloring of $G$ if for every vertex $v\in V$, at least half of the vertices in the closed neighborhood $N[v]$ of $v$ have the same color as $v$. The maximum cardinality of a quorum coloring of $G$ is called the quorum coloring number of $G$ and is denoted by $\psi_{q}(G)$. A quorum coloring of order $\psi_{q}(G)$ is a $\psi_{q}$-coloring. The determination of the quorum coloring number or design a linear-time algorithm computing it in a perfect $N$-ary tree has been posed recently as an open problem by Sahbi. In this paper, we answer this problem by designing a linear-time algorithm for finding both a $\psi_{q}$-coloring and the quorum coloring number for every perfect tree whose the vertices at the same depth have the same degree.

Abstract: Each graphon $W:\Omega^2\rightarrow[0,1]$ yields an inhomogeneous random graph model $G(n,W)$. We show that $G(n,W)$ is asymptotically almost surely connected if and only if (i) $W$ is a connected graphon and (ii) the measure of elements of $\Omega$ of $W$-degree less than $\alpha$ is $o(\alpha)$ as $\alpha\rightarrow 0$. These two conditions encapsulate the absence of several linear-sized components, and of isolated vertices, respectively. We study in bigger detail the limit probability of the property that $G(n,W)$ contains an isolated vertex, and, more generally, the limit distribution of the minimum degree of $G(n,W)$.

Abstract: A minimum storage regenerating (MSR) subspace family of $\mathbb{F}_q^{2m}$ is a set $\mathcal{S}$ of $m$-spaces in $\mathbb{F}_q^{2m}$ such that for any $m$-space $S$ in $\mathcal{S}$ there exists an element in $\mathrm{PGL}(2m, q)$ which maps $S$ to itself and fixes $\mathcal{S} \setminus \{ S \}$ pointwise. We show that an MSR subspace family of $2$-spaces in $\mathbb{F}_q^4$ has at most size $6$ with equality if and only if it is a particular subset of a Segre variety. This implies that an $(n, n-2, 4)$-MSR code has $n \leq 9$.

Abstract: Graphical parking functions, or $G$-parking functions, are a generalization of classical parking functions which depend on a connected multigraph $G$ having a distinguished root vertex. Gaydarov and Hopkins characterized the relationship between $G$-parking functions and another vector-dependent generalization of parking functions, the $\boldsymbol{u}$-parking functions. The crucial component of their result was their classification of all graphs $G$ whose $G$-parking functions are invariant under action by the symmetric group $\mathfrak{S}_n$, where $n+1$ is the order of $G$. In this work, we present a 2-dimensional analogue of Gaydarov and Hopkins' results by characterizing the overlap between $G$-parking functions and 2-dimensional $\boldsymbol{U}$-parking functions, i.e., pairs of integer sequences whose order statistics are bounded by certain weights along lattice paths in the plane. Our key result is a total classification of all $G$ whose set of $G$-parking functions is $(\mathfrak{S}_p \times \mathfrak{S}_q)$-invariant, where $p+q+1$ is the order of $G$.