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This paper deals with the problem of finding totally antimagic total labelings of complete bipartite graphs. We prove that complete bipartite graphs are totally antimagic total graphs. |
In 2009, Bang-Jensen asked whether there exists a function $g(k)$ such that every strongly $k$-connected $n$-vertex tournament contains a strongly $k$-connected spanning subgraph with at most $kn + g(k)$ arcs. In this paper, we answer the question by showing that every strongly $k$-connected $n$-vertex tournament cont... |
We give an elementary and self-contained proof, and a numerical improvement, of a weaker form of the excluded clique minor theorem of Robertson and Seymour, the following. Let t,r>0 be integers, and let R=49152t^{24}(40t^2+r). |
The notion of palindromic length of a finite word, as well as an infinite word, was first introduced by Frid, Puzynina and Zamboni\cite{FRID2013737}. They conjectured that if the palindromic length of an infinite word is bounded, then this sequence is eventually periodic. |
Terao's factorization theorem shows that if an arrangement is free, then its characteristic polynomial factors into the product of linear polynomials over the integer ring. This is not a necessary condition, but there are not so many non-free arrangements whose characteristic polynomial factors over the integer ri... |
In this paper we prove that the square of an essentially 2-edge connected graph with an additional property has a connected even factor with maximum degree at most 4. Moreover we show that, in general, the square of essentially 2-edge connected graph does not contain a connected even factor with bounded maximum degree... |
The Ehrhart quasipolynomial of a rational polytope $\mathsf{P}$ encodes fundamental arithmetic data of $\mathsf{P}$, namely, the number of integer lattice points in positive integral dilates of $\mathsf{P}$. Ehrhart quasipolynomials were introduced in the 1960s, satisfy several fundamental structural results and have ... |
A graph is properly edge-colored if no two adjacent edges have the same color. The smallest number of edges in a graph any of whose proper edge colorings contains a totally multicolored copy of a graph H is the size anti-Ramsey number AR_s(H) of H. This number in offline and online setting is investigated here. |
We study metric properties of the cone of homogeneous non-negative multivariate polynomials and the cone of sums of powers of linear forms, and the relationship between the two cones. We compute the maximum volume ellipsoid of the natural base of the cone of non-negative polynomials and the minimum volume ellipsoid of... |
We are given a set $A$ of buyers, a set $B$ of houses, and for each buyer a preference list, i.e., an ordering of the houses. A house allocation is an injective mapping $\tau$ from $A$ to $B$, and $\tau$ is strictly better than another house allocation $\tau'\neq \tau$ if for every buyer $i$, $\tau'(i)$ does n... |
We introduce a new model of random multigraphs with colored vertices and weighted edges. It is similar to the "inhomogeneous random graph model" of Söderberg (2002), extended by Bollobás, Janson and Riordan (2007). |
The reconfiguration graph $\mathcal{C}_k(G)$ for the $k$-colourings of a graph $G$ has a vertex for each proper $k$-colouring of $G$, and two vertices of $\mathcal{C}_k(G)$ are adjacent precisely when those $k$-colourings differ on a single vertex of $G$. Much work has focused on bounding the maximum value of ${\rm{di... |
The study of uniform Turán densities was initiated in the 1980s by Erdős and Sós. Given a $3$-graph $F$, the uniform Turán density of $F$, $\pi_{\therefore}(F)$, is defined as the infimum $d\in[0,1]$ such that every $3$-graph $H$ in which every linearly sized $S\subseteq V(H)$ induces at least $(d+o(1))\binom{\vert S\... |
The packing number of a graph $G$ is the maximum number of closed neighborhoods of vertices in $G$ with pairwise empty intersections. Similarly, the open packing number of $G$ is the maximum number of open neighborhoods in $G$ with pairwise empty intersections. |
The Kruskal-Katona theorem together with a theorem of Razborov determine the closure of the set of points defined by the homomorphism density of the edge and the triangle in finite graphs. The boundary of this region is a countable union of algebraic curves, and in particular, it is almost everywhere differentiable. |
Answering a question of P. Erdos from 1965, we show that for every eps>0 there is a set A of n integers with the following property: every subset A' of A with at least (1/3 + eps)n elements contains three distinct elements x,y,z with x + y = z. |
Let $G$ be a simple graph with maximum degree denoted as $\Delta(G)$. An overfull subgraph $H$ of $G$ is a subgraph satisfying the condition $|E(H)| > \Delta(G)\lfloor \frac{1}{2}|V(H)| \rfloor$. |
Let $D = d_1, d_2, \ldots, d_n$ and $F = f_1, f_2,\ldots, f_n$ be two sequences of positive integers. We consider the following decision problems: is there a $i)$ multigraph, $ii)$ loopless multigraph, $iii)$ simple graph, $iv)$ connected simple graph, $v)$ tree, $vi)$ caterpillar $G = (V,E)$ such that for all $k$, $d... |
Let J and J* be subsets of Z+ such that 0,1\in J and 0\in J*. For infinitely many n, let k=(k_1,..., k_n) be a vector of nonnegative integers whose sum M is even. |
We present an example of a symmetric tensor of size $800\times 800\times 800$ which can be written a sum of $903$ simple tensors with complex entries but not as a sum of $903$ symmetric simple tensors. |
We propose a class of chess variants, Multimove Chess (i,j), in which White gets i moves per turn and Black gets j moves per turn. One side is said to win when it takes the opponent's king. |
We prove that for all natural numbers $m$ and $k$ where $k$ is odd, there exists a natural number $N(k)$ such that any 3-connected cubic graph with at least $N(k)$ vertices contains a cycle of length $m$ modulo $k$. We also construct a family of graphs showing that this is not true for 2-connected cubic graphs if $m$ ... |
A 2-hued coloring of a graph $G$ (also known as conditional $(k, 2)$-coloring and dynamic coloring) is a coloring such that for every vertex $v\in V(G)$ of degree at least $2$, the neighbors of $v$ receive at least $2$ colors. The smallest integer $k$ such that $G$ has a 2-hued coloring with $ k $ colors, is called th... |
The translational tiling problem, dated back to Wang's domino problem in the 1960s, is one of the most representative undecidable problems in the field of discrete geometry and combinatorics. Ollinger initiated the study of the undecidability of translational tiling with a fixed number of tiles in 2009, and proved... |
Consider a matrix $M$ chosen uniformly at random from a class of $m \times n$ matrices of zeros and ones with prescribed row and column sums. A partially filled matrix $D$ is a $\mathit{defining}$ $\mathit{set}$ for $M$ if $M$ is the unique member of its class that contains the entries in $D$. |
The purpose of this note is to complete the study, begun in the first author's PhD thesis, of the topology of the poset of generalized noncrossing partitions associated to real reflection groups. In particular, we calculate the Euler characteristic of this poset with the maximal and minimal elements deleted. |
By using biclique partitions of digraphs, this paper gives reduction formulas for the number of oriented spanning trees, stationary distribution vector and Kemeny's constant of digraphs. As applications, we give a method for enumerating spanning trees of undirected graphs by vertex degrees and biclique partitions.... |
The well-known notion of domination in a graph abstracts the idea of protecting locations with guards. This paper introduces a new graph invariant, the autonomous domination number, which abstracts the idea of defending a collection of locations with autonomous agents following a simple protocol to coordinate their de... |
We give historical remarks related to <a href="https://arxiv.org/abs/2112.14547" data-arxiv-id="2112.14547" class="link-https">arXiv:2112.14547</a> ("A New Method of Construction of Permutation Trinomials with Coefficients 1", by Guo et al.). In particular, we show that the "new" permutation polynomial... |
We give yet another proof for Faà di Bruno's formula for higher derivatives of composite functions. Our proof technique relies on reinterpreting the composition of two power series as the generating function for weighted integer compositions, for which a Faà di Bruno-like formula is quite naturally established. |
Aguiar and Mahajan provided several coalgebra axioms and algebra axioms for a family of left-regular bands to construct a commutative diagram of algebras and coalgebras. In this paper, we will add compatibility axioms to make it a diagram of Hopf algebras. |
We establish splitter theorems for graph immersions for two families of graphs, $k$-edge-connected graphs, with $k$ even, and 3-edge-connected, internally 4-edge-connected graphs. As a corollary, we prove that every $3$-edge-connected, internally $4$-edge-connected graph on at least seven vertices that immerses $K_5$ ... |
A path in an edge-colored graph is called {\em rainbow} if no two edges of it are colored the same. For an $\ell$-connected graph $G$ and an integer $k$ with $1\leq k\leq \ell$, the {\em rainbow $k$-connection number} $rc_k(G)$ of $G$ is defined to be the minimum number of colors required to color the edges of $G$ suc... |
A pebbling move on a graph removes two pebbles at a vertex and adds one pebble at an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. |
An improper interval (edge) coloring of a graph $G$ is an assignment of colors to the edges of $G$ satisfying the condition that, for every vertex $v \in V(G)$, the set of colors assigned to the edges incident with $v$ forms an integral interval. An interval coloring is $k$-improper if at most $k$ edges with the same ... |
We count numbers of tame frieze patterns with entries in a finite commutative local ring. For the ring $\mathbb{Z}/p^r\mathbb{Z}$, $p$ a prime and $r\in\mathbb{N}$ we obtain closed formulae for all heights. |
In their study of cyclic pattern containment, Domagalski et al. conjecture differential equations for the generating functions of circular permutations avoiding consecutive patterns of length 3. In this note, we prove and significantly generalize these conjectures. |
Menger's well-known theorem from 1927 characterizes when it is possible to find $k$ vertex-disjoint paths between two sets of vertices in a graph $G$. Recently, Georgakopoulos and Papasoglu and, independently, Albrechtsen, Huynh, Jacobs, Knappe and Wollan conjectured a coarse analogue of Menger's theorem, when... |
Determining the maximum size of a $t$-intersecting code in $[m]^n$ was a longstanding open problem of Frankl and Füredi, solved independently by Ahlswede and Khachatrian and by Frankl and Tokushige. We extend their result to the setting of forbidden intersections, by showing that for any $m>2$ and $n$ large compare... |
In this paper, we study two different subposets of the $\nu$-Tamari lattice: one in which all elements have maximal in-degree and one in which all elements have maximal out-degree. The maximal in-degree and maximal out-degree of a $\nu$-Dyck path turns out to be the size of the maximal staircase shape path that fits w... |
In [Graphs Combin.~24 (2008) 469--483.], the third author and the fifth author conjectured that if $G$ is a $k$-connected graph such that $\sigma_{k+1}(G) \ge |V(G)|+\kappa(G)+(k-2)(\alpha(G)-1)$, then $G$ contains a Hamiltonian cycle, where $\sigma_{k+1}(G)$, $\kappa(G)$ and $\alpha(G)$ are the minimum degree sum of $... |
An edge-colored hypergraph is rainbow if all of its edges have different colors. Given two hypergraphs $\mathcal{H}$ and $\mathcal{G}$, the anti-Ramsey number $ar(\mathcal{G}, \mathcal{H})$ of $\mathcal{H}$ in $\mathcal{G}$ is the maximum number of colors needed to color the edges of $\mathcal{G}$ so that there does n... |
Let $G$ be a connected graph of order $n$, $F_k$ be a fan consisting of $k$ triangles sharing a common vertex, and $tF_k$ be $t$ vertex-disjoint copies of $F_k$. Brennan (2017) showed the Ramsey number $r(G,F_k)=2n-1$ for $G$ being a unicyclic graph for $n \geq k^2-k+1$ and $k\ge 18$, and asked the threshold $c(n)$ fo... |
A graph $G$ is \textit{$k$-critical} if $\chi(G) = k$ and every proper subgraph of $G$ is $(k - 1)$-colorable, and if $L$ is a list-assignment for $G$, then $G$ is \textit{$L$-critical} if $G$ is not $L$-colorable but every proper induced subgraph of $G$ is. In 2014, Kostochka and Yancey proved a lower bound on the av... |
Let $G$ be a connected simple graph of order $n$. Let $\rho_1(G)\geq \rho_2(G)\geq \cdots \geq \rho_{n-1}(G)> \rho_n(G)=0$ be the eigenvalues of the normalized Laplacian matrix $\mathcal{L}(G)$ of $G$. |
The distance of a vertex in a graph is the sum of distances from that vertex to all other vertices of the graph. The Wiener index of a graph is the sum of distances between all its unordered pairs of vertices. |
For any odd $t\ge 9$, we present a polynomial-time algorithm that solves the $3$-colouring problem, and finds a $3$-colouring if one exists, in $P_{t}$-free graphs of odd girth at least $t-2$. In particular, our algorithm works for $(P_9, C_3, C_5)$-free graphs, thus making progress towards determining the complexity ... |
Albertson defined the irregularity of a graph $G$ as $irr(G)=\sum\limits_{uv\in E(G)}|d_G (u)-d_G |
Let S(n) be the integer sequence which is the coefficient of x^{n(n+1)/4} in the expansion of (1+x)(1+x^2), ..., (1+x^n) for positive integers n congruent to 0 or 3 mod 4. We prove a conjecture of Andrica and Tomescu that S(n) is asymptotic to \sqrt{6/\pi} 2^n n^{-3/2} as n approaches infinity. |
Ellenberg and Gijswijt gave the best known asymptotic upper bound for the cardinality of subsets of $\mathbb F_q^n$ without 3-term arithmetic progressions. We improve this bound by a factor $\sqrt{n}$. |
For a labelled tree on the vertex set $[n]:=\{1,2,..., n\}$, define the direction of each edge $ij$ to be $i\to j$ if $i<j$. The indegree sequence of $T$ can be considered as a partition $\lambda \vdash n-1$. |
The Maker-Breaker resolving game is a game played on a graph $G$ by Resolver and Spoiler. The players taking turns alternately in which each player selects a not yet played vertex of $G$. |
The Turán number of a graph H, ex(n;H), is the maximum number of edges in any graph on n vertices which does not contain H as a subgraph. Let P_l denote a path on l vertices, and kP_l denote k vertex-disjoint copies of P_l. |
We prove that if Cay(G;S) is a connected Cayley graph with n vertices, and the prime factorization of n is very small, then Cay(G;S) has a hamiltonian cycle. More precisely, if p, q, and r are distinct primes, then n can be of the form kp with k < 32 and k not equal to 24, or of the form kpq with k < 6, or of th... |
A permutation sequence is said to be convergent if the density of occurrences of every fixed permutation in the elements of the sequence converges. We prove that such a convergent sequence has a natural limit object, namely a Lebesgue measurable function $Z:[0,1]^2 \to [0,1]$ with the additional properties that, for e... |
Coincidence site lattices of oblique planar lattices are algebraically characterized using as basic tool the Cartan-Dieudonné theorem, that is, the decomposition of an orthogonal transformation as a product of reflections. The case of rectangular lattices is worked out in detail. |
Let $F$ be a graph, and let $\mathcal{B}_r(F)$ be the class of $r$-uniform Berge-$F$ hypergraphs. In this paper, we establish a relationship between the spectral radius of the adjacency tensor of a uniform hypergraph and its local structure through walks. |
Shareshian-Wachs, Brosnan-Chow, and Guay-Pacquet [Adv. Math. |
The connected domination game is a variation of the domination game where the played vertices must form a connected subgraph at all stages of the game. In this paper we prove that deciding whether the game connected domination number is smaller than a given integer is PSPACE-complete using log-space reductions for bot... |
A binary word is called $q$-decreasing, for $q>0$, if inside this word each of length-maximal (in the local sense) occurrences of a factor of the form $0^a1^b$, $a>0$, satisfies $q \cdot a > b$. We bijectively link $q$-decreasing words with certain prefixes of the cutting sequence of the line $y=qx$. |
Jim Propp's rotor router model is a deterministic analogue of a random walk on a graph. Instead of distributing chips randomly, each vertex serves its neighbors in a fixed order. |
Motivated by some questions in Euclidean Ramsey theory, our aim in this note is to show that there exists a cyclic quadrilateral that does not embed into any transitive set (in any dimension). We show that in fact this holds for almost all cyclic quadrilaterals, and we also give explicit examples of such cyclic quadri... |
The Eulerian polynomials and derangement polynomials are two well-studied generating functions that frequently arise in combinatorics, algebra, and geometry. When one makes an appearance, the other often does so as well, and their corresponding generalizations are similarly linked. |
A nonnegative multidimensional matrix is called polystochastic if the sum of its entries over each line is equal to $1$. In this paper we overview known results on positiveness of the permanent of polystochastic matrices and prove that the permanent of every $4$-dimensional polystochastic matrix of order $4$ is greate... |
In this paper, we prove the asymptotic multipartite version of the Alon-Yuster theorem, which is a generalization of the Hajnal-Szemerédi theorem: If $k\geq 3$ is an integer, $H$ is a $k$-colorable graph and $\gamma>0$ is fixed, then, for every sufficiently large $n$, where $|V(H)|$ divides $n$, and for every balanc... |
In 2014, Cégielski, Grigorieff and Guessarian characterized unary self-maps on the set $\mathbb{Z}$ of integers which preserve all congruences of the additive group. In this note, we propose a shorter and straigthforward proof. |
The concept of super line graph was introduced in the year 1995 by Bagga, Beineke and Varma. Given a graph with at least $r$ edges, the super line graph of index $r$, $L_r(G)$, has as its vertices the sets of $r$ edges of $G$, with two adjacent if there is an edge in one set adjacent to an edge in the other set. |
We define the notion of infinite friezes of positive integers as a variation of Conway-Coxeter frieze patterns and study their properties. We introduce useful gluing and cutting operations on infinite friezes. |
We describe an algorithm to compute the geodesics in an arbitrary CAT(0) cubical complex. A key tool is a correspondence between cubical complexes of global non-positive curvature and posets with inconsistent pairs. |
We describe here how the recent Wagner's approach for applying reinforcement learning to construct examples in graph theory can be used in the search for critical graphs for small Ramsey numbers. We illustrate this application by providing lower bounds for the small Ramsey numbers $R(K_{2,5}, K_{3,5})$, $R(B_3, B_... |
We study two matrices $N$ and $M$ defined by the parameters of equivalent $S$- and $J$-continued fraction expansions, and compare them by examining the product $N^{-1}M$. Using examples based on the Catalan numbers, the little Schröder numbers and powers of $q$, we indicate that this matrix product is an object worthy... |
For a fixed graph $H$, we say that an edge-colored graph $G$ is \emph{weakly $H$-rainbow saturated} if there exists an ordering $e_1, e_2, \ldots, e_m$ of $E\left(\overline{G}\right)$ such that, for any list $c_1, c_2, \ldots, c_m$ of pairwise distinct colors from $\mathbb{N}$, the non-edges $e_i$ in color $c_i$ can be... |
The concept of sequency holds a fundamental significance in signal analysis using Walsh basis functions. In this study, we closely examine the concept of sequency and explore the properties of sequency-complete and sequency-ordered matrices. |
In this note we construct a flag simplicial $3$-sphere $\Delta$ with the following properties: - $\Delta$ is not a suspension; - $\Delta$ has no edge that can be contracted to obtain another flag sphere; - The only equators (induced subcomplexes which are spheres of codimension $1$) of $\Delta$ are vertex links. Our c... |
In this paper, we try to minimize the scope of possible unique metric spectra up to equivalence. While it is well known that every spectra $S\subseteq \mathbb{R}^+$ is equivalent to a spectra $T\subseteq \mathbb{N}$, it has remained open if $T$ could also maintain a desirable combinatorial form. |
We show that the d-cube is determined by the spectrum of its distance matrix. |
The clique graph $kG$ of a graph $G$ has as its vertices the cliques (maximal complete subgraphs) of $G$, two of which are adjacent in $kG$ if they have non-empty intersection in $G$. We say that $G$ is clique convergent if $k^nG\cong k^m G$ for some $n\not= m$, and that $G$ is clique divergent otherwise. |
We use the fact that certain cosets of the stabilizer of points are pairwise conjugate in a symmetric group $S_n$ in order to construct recurrence relations for enumerating certain subsets of $S_n$. Occasionally one can find `closed form' solutions to such recurrence relations. |
For $k\ge 3$ and $\epsilon>0$, let $H$ be a $k$-partite $k$-graph with parts $V_1,\dots, V_k$ each of size $n$, where $n$ is sufficiently large. Assume that for each $i\in [k]$, every $(k-1)$-set in $\prod_{j\in [k]\setminus \{i\}} V_i$ lies in at least $a_i$ edges, and $a_1\ge a_2\ge \cdots \ge a_k$. |
In this paper we establish an improved bound for the number of incidences between a set $P$ of $m$ points and a set $H$ of $n$ planes in $\mathbb R^3$, provided that the points lie on a two-dimensional nonlinear irreducible algebraic variety $V$ of constant degree. Specifically, the bound is $$ O\left( m^{2/3}n^{2/3} ... |
In an earlier paper the authors proved that limits of convergent graph sequences can be described by various structures, including certain 2-variable real functions called graphons, random graph models satisfying certain consistency conditions, and normalized, multiplicative and reflection positive graph parameters. I... |
Bishop independence concerns determining the maximum number of bishops that can be placed on a board such that no bishop can attack any other bishop. This paper presents the solution to the bishop independence problem, determining the bishop independence number, for all sizes of boards on the surface of a square prism... |
In this paper, we consider the problem of reducing the semitotal domination number of a given graph by contracting $k$ edges, for some fixed $k \geq 1$. We show that this can always be done with at most 3 edge contractions and further characterise those graphs requiring 1, 2 or 3 edge contractions, respectively, to de... |
We describe a new construction algorithm for the recursive generation of all non-isomorphic IPR fullerenes. Unlike previous algorithms, the new algorithm stays entirely within the class of IPR fullerenes, that is: every IPR fullerene is constructed by expanding a smaller IPR fullerene unless it belongs to limited clas... |
We construct a simple acyclic directed graph for which the Bunkbed Conjecture is false, thereby resolving conjectures posed by Leander and by Hollom. |
For a graph $\Gamma=(V\Gamma,E\Gamma)$, a subset $D$ of $V\Gamma$ is a perfect code in $\Gamma$ if every vertex of $\Gamma$ is dominated by exactly one vertex in $D$. In this paper, we classify all connected quartic Cayley graphs on generalized dihedral groups admitting a perfect code, and determine all perfect codes ... |
We study a generalisation of the bipartite Ramsey numbers to blowups of graphs. For a graph $G$, denote the $t$-blowup of $G$ by $G[t]$. |
Let $\pi$ be an order-$q$-subplane of $PG(2,q^3)$ that is exterior to $\ell_\infty$. Then the exterior splash of $\pi$ is the set of $q^2+q+1$ points on $\ell_\infty$ that lie on an extended line of $\pi$. |
Faults and viruses often spread in networked environments by propagating from site to neighboring site. We model this process of {\em network contamination} by graphs. |
The concepts of symmetric and asymmetric peaks in Dyck paths were introduced by Flórez and Ram\'ırez, who counted the total number of such peaks over all Dyck paths of a given length. Elizalde generalized their results by giving multivariate generating functions that keep track of the number of symmetric peaks and... |
In this paper, we first establish the very close link between stability of graphs, a concept first introduced in \cite{Scapsalvi1} and studied most notably by Surowski \cite{Surowski1}, \cite{Surowski2} and Wilson \cite{Wilson01} and two-fold automorphisms. The concept of two-fold isomorphisms, as far as we know, firs... |
We introduce Row Impartial Terminus (RIT), an impartial combinatorial game played on integer partitions. We show that any position in RIT can be uniquely decomposed into a core and a remnant. |
Friezes with coefficients are maps assigning numbers to the edges and diagonals of a regular polygon such that all Ptolemy relations for crossing diagonals are satisfied. Among these, the classic Conway-Coxeter friezes are the ones where all values are natural numbers and all edges have value 1. |
If F(x) = e^G(x), where F(x) = \sum f(n)x^n and $G(x) = \sum g(n)x^n, with 0 \le g(n) = O(n^{theta n}/n!),theta in (0,1), and gcd(n : g(n) > 0)=1, then f(n) = o(f(n-1)). <br>This gives an answer to Compton's request in Question 8.3 for an ``easily verifiable sufficient condition'' to show that an adequa... |
In this paper we describe a triple correspondence between graph limits, information theory and group theory. We put forward a new graph limit concept called log-convergence that is closely connected to dense graph limits but its main applications are in the study of sparse graph sequences. |
We define an algebraic setup of homology for hypergraphs, which defaults to simplicial homology in the case of graphs, and study its basic properties. As part of our study we define algebraic spanning trees of hypergraphs, along with fundamental cuts and cycles playing their usual roles. |
In 2002 Zhi-Wei Sun [Integers 2(2002)] published a curious identity involving binomial coefficients. In this paper we present a generalization of the identity. |
Denote by $\mathcal{C}^-_{\ell}$ the $3$-uniform hypergraph obtained by removing one hyperedge from the tight cycle on $\ell$ vertices. It is conjectured that the Turán density of $\mathcal{C}^-_{5}$ is $1/4$. |
The chordal ring (CR) graphs are a well-known family of graphs used to model some interconnection networks for computer systems in which all nodes are in a cycle. Generalizing the CR graphs, in this paper, we introduce the families of chordal multi-ring (CMR), chordal ring mixed (CRM), and chordal multi-ring mixed (CM... |
To any infinite word w over a finite alphabet A we can associate two infinite words min(w) and max(w) such that any prefix of min(w) (resp. max(w)) is the lexicographically smallest (resp. greatest) amongst the factors of w of the same length. |
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