Combinatorics (math.CO)

Abstract: In this paper, we determine the graphs which have the minimal spectral radius among all the connected graphs of order $n$ and the independence number $\lceil\frac{n}{2}\rceil-1.$

Abstract: In this paper, we obtain asymptotic formulas for $k$-crank of $k$-colored partitions. Let $M_k(a, c; n)$ denote the number of $k$-colored partitions of $n$ with a $k$-crank congruent to $a$ mod $c$. For the cases $k=2,3,4$, Fu and Tang derived several inequality relations for $M_k(a, c; n)$ using generating functions. We employ the Hardy-Ramanujan Circle Method to extend the results of Fu and Tang. Furthermore, additional inequality relations for $M_k(a, c; n)$ have been established, such as logarithmic concavity and logarithmic subadditivity.

Abstract: In this paper, we study the connected blocks polytope, which, apart from its own merits, can be seen as the generalization of certain connectivity based or Eulerian subgraph polytopes. We provide a complete facet description of this polytope, characterize its edges and show that it is Hirsch. We also show that connected blocks polytopes admit a regular unimodular triangulation by constructing a squarefree Gr\"obner basis. In addition, we prove that the polytope is Gorenstein of index $2$ and that its $h^\ast$-vector is unimodal.

Abstract: In this paper we use the Lyndon-shirshov basis to study the shuffle type polynomials. We give a free noncommutative binomial (or multinomial) theorem in terms of the Lyndon-Shirshov basis. Another noncommutative binomial theorem given by the shuffle type polynomials with respect to an adjoint derivation is established. As a result, the Bell differential polynomials and the $q$-Bell differential polynomials can be derived from the second binomial theorem. The relation between the shuffle type polynomials and the Bell differential polynomials is established. Finally, we give some applications of the free noncommutative binomial theorem including application of the shuffle type polynomials to bialgebras and Hopf algebras.

Abstract: Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. To be more formal, a graph $G$ is a median graph if, for all $\mu, u,v\in V(G)$, it holds that $|I(\mu,u)\cap I(\mu,v)\cap I(u,v)|=1$ where $I(x,y)$ denotes the set of all vertices that lie on shortest paths connecting $x$ and $y$. In this paper we are interested in a natural generalization of median graphs, called $k$-median graphs. A graph $G$ is a $k$-median graph, if there are $k$ vertices $\mu_1,\dots,\mu_k\in V(G)$ such that, for all $u,v\in V(G)$, it holds that $|I(\mu_i,u)\cap I(\mu_i,v)\cap I(u,v)|=1$, $1\leq i\leq k$. By definition, every median graph with $n$ vertices is an $n$-median graph. We provide several characterizations of $k$-median graphs that, in turn, are used to provide many novel characterizations of median graphs.

Abstract: Given integer $n$ and $k$ such that $0 < k \leq n$ and $n$ piles of stones, two player alternate turns. By one move it is allowed to choose any $k$ piles and remove exactly one stone from each. The player who has to move but cannot is the loser. Cases $k=1$ and $k = n$ are trivial. For $k=2$ the game was solved for $n \leq 6$. For $n \leq 4$ the Sprague-Grundy function was efficiently computed (for both the normal and mis\`ere versions). For $n = 5,6$ a polynomial algorithm computing P-positions was obtained. Here we consider the case $2 \leq k = n-1$ and compute Smith's remoteness function, whose even values define the P-positions. In fact, an optimal move is always defined by the following simple rule: if all piles are odd, keep a largest one and reduce all other; if there exist even piles, keep a smallest one of them and reduce all other. Such strategy is optimal for both players, moreover, it allows to win as fast as possible from an N-position and to resist as long as possible from a P-position.

Abstract: A permutation is said to be cycle-alternating if it has no cycle double rises, cycle double falls or fixed points; thus each index $i$ is either a cycle valley ($\sigma^{-1}(i)>i<\sigma(i)$) or a cycle peak ($\sigma^{-1}(i)<i>\sigma(i)$). We find Stieltjes-type continued fractions for some multivariate polynomials that enumerate cycle-alternating permutations with respect to a large (sometimes infinite) number of simultaneous statistics that measure cycle status, record status, crossings and nestings along with the parity of the indices. Our continued fractions are specializations of more general continued fractions of Sokal and Zeng. We then introduce alternating Laguerre digraphs, which are generalization of cycle-alternating permutations, and find exponential generating functions for some polynomials enumerating them. We interpret the Stieltjes--Rogers and Jacobi--Rogers matrices associated to some of our continued fractions in terms of alternating Laguerre digraphs.

Abstract: The Queen's Domination problem, studied for over 160 years, poses the following question: What is the least number of queens that can be arranged on a $m \times n$ chessboard so that they either attack or occupy every cell? We propose a novel relaxation of the Queen's Domination problem and show that it is exactly solvable on both square and rectangular chessboards. As a consequence, we improve on the best known lower bound for rectangular chessboards in $\approx 12.5\%$ of the non-trivial cases. As another consequence, we simplify and generalize the proofs for the best known lower-bounds for Queen's Domination of square $n \times n$ chessboards for $n \equiv \{0,1,2\} \mod 4$ using an elegant idea based on a convex hull. Finally, we show some results and make some conjectures towards the goal of simplifying the long complicated proof for the best known lower-bound for square boards when $n \equiv 3 \mod 4$ (and $n > 11$). These simple-to-state conjectures may also be of independent interest.

Abstract: For each integer partition $\lambda \vdash n$ we give a simple combinatorial expression for the sum of the Jack character $\theta^\lambda_\alpha$ over the integer partitions of $n$ with no singleton parts. For $\alpha = 1,2$ this gives closed forms for the eigenvalues of the permutation and perfect matching derangement graphs, resolving an open question in algebraic graph theory. A byproduct of the latter is a simple combinatorial formula for the immanants of the matrix $J-I$ where $J$ is the all-ones matrix, which might be of independent interest. Our proofs center around a Jack analogue of a hook product related to Cayley's $\Omega$--process in classical invariant theory, which we call the principal lower hook product.

Abstract: Let $G$ be a multigraph. A subset $F$ of $E(G)$ is an edge cover of $G$ if every vertex of $G$ is incident to an edge of $F$. The cover index, $\xi(G)$, is the largest number of edge covers into which the edges of $G$ can be partitioned. Clearly $\xi(G) \le \delta(G)$, the minimum degree of $G$. For $U\subseteq V(G)$, denote by $E^+(U)$ the set of edges incident to a vertex of $U$. When $|U|$ is odd, to cover all the vertices of $U$, any edge cover needs to contain at least $(|U|+1)/2$ edges from $E^+(U)$, indicating $ \xi(G) \le |E^+(U)|/ (|U|+1)/2$. Let $\rho_c(G)$, the co-density of $G$, be defined as the minimum of $|E^+(U)|/((|U|+1)/2)$ ranging over all $U\subseteq V(G)$ with $|U| $ odd and at least 3. Then $\rho_c(G)$ provides another upper bound on $\xi(G)$. Thus $\xi(G) \le \min\{\delta(G), \lfloor \rho_c(G) \rfloor \}$. For a lower bound on $\xi(G)$, in 1967, Gupta conjectured that $\xi(G) \ge \min\{\delta(G)-1, \lfloor \rho_c(G) \rfloor \}$. Gupta showed that the conjecture is true when $G$ is simple, and Cao et al. verified this conjecture when $\rho_c(G)$ is not an integer. In this note, we confirm the conjecture when the maximum multiplicity of $G$ is at most two or $ \min\{\delta(G)-1, \lfloor \rho_c(G) \rfloor \} \le 6$.