Let $K_{j\times n}$ denote a complete multipartite graph with $j$ partite sets, each containing $n$ vertices. For given graphs $G_1$ and $G_2$, the multipartite Ramsey number $m_j(G_1,G_2)$ is the smallest integer $t$ such that any subgraph $G$ of $K_{j\times t}$ either contains a copy of $G_1$, or its complement relative to $K_{j\times t}$ contains a copy of $G_2$. The exact values of $m_j(nK_2,C_m)$ are known for several values of $j$ and $m$. The problem is fully resolved for $m\leq 6$, while for $m=7$ the exact values were previously known for $j\leq 4$. In this article, we determine $m_j(nK_2,C_7)$ for $j\geq 5$ and $n\geq 2$, completing the remaining cases for $m=7$.
Let $\mathcal{C}_n$ denote the set of Catalan words of length $n$ and $\mathcal{C}_n(p)$ the subset of $\mathcal{C}_n$ whose members avoid the pattern $p$. In this paper, we enumerate members of $\mathcal{C}_n(p)$ where $p$ is a permutation pattern of length four. This extends recent work on Catalan words concerning the avoidance of classical, consecutive or vincular patterns of length at most three. Indeed, we determine the generating function of the distribution for the number of descents on each of the corresponding avoidance classes. We make use of the symbolic counting method to establish our results, together with a variety of other enumerative techniques, including use of strategic decompositions, introduction of auxiliary generating functions and analysis of active sites.
We provide new constructions of Meyniel extremal graphs, which are families of graphs with the conjectured largest asymptotic cop number. Using spanning subgraphs, we prove that there are an exponential number of new Meyniel extremal families with specified degrees. Using linear programming on hypergraphs, we explore the degrees in families that are not Meyniel extremal. We give the best-known upper bound on the cop number of vertex-transitive graphs with a prescribed degree. We find new Meyniel extremal families of regular graphs with large chromatic number, large diameter, and explore the connection between Meyniel extremal graphs and bipartite graphs. Conjectures relating Meyniel extremal families to maximum and average degrees in their graphs are presented.
In this note we determine the series expansion of the logarithm of the exponential generating function of Eulerian polynomials, which results in a new identity on Eulerian polynomials. We also obtain similar results for general Eulerian polynomials introduced by Xiong, Tsao, and Hall. As consequences, we derive some relations between classical Eulerian polynomials and their variations.
This paper studies three generalized q-series combinatorially using split (n + t)-color partitions as a combinatorial tool. This work provides a generalized approach to unify the several combinatorial identities found in the literature. In this process, we obtain several new Rogers-Ramanujan-MacMahon type partition identities.
