Some multicolor bipartite Ramsey numbers involving cycles and a small number of colors

For bipartite graphs $G_1,G_2,\dots ,G_k$, the bipartite Ramsey number $b(G_1,G_2,\dots ,G_k)$ is the least positive integer $b$ so that any coloring of the edges of $K_{b,b}$ with $k$ colors will result in a copy of $G_i$ in the $i$th color for some $i$. In this paper, our main focus will be to bound the following numbers: $b(C_{2t_1},C_{2t_2})$ and $b(C_{2t_1},C_{2t_2},C_{2t_3})$ for all $t_i\ge 3,$$b(C_{2t_1},C_{2t_2},C_{2t_3},C_{2t_4})$ for $3\le t_i\le 9,$ and $b(C_{2t_1},C_{2t_2},C_{2t_3},C_{2t_4},C_{2t_5})$ for $3\le t_i\le 5.$ Furthermore, we will also show that these mentioned bounds are generally better than the bounds obtained by using the best known Zarankiewicz-type result.     

Embeddings of rank-2 tori in algebraic groups

Let k be a field of characteristic different from 2 and 3. In this paper we study connected simple algebraic groups of type $A_2$, $G_2$ and $F_4$ defined over k, via their rank-2 k-tori. Simple, simply connected groups of type $A_2$ play a pivotal role in the study of exceptional groups and this aspect is brought out by the results in this paper. We refer to tori, which are maximal tori of $A_n$ type groups, as unitary tori. We discuss conditions necessary for a rank-2 unitary k-torus to embed in simple k-groups of type $A_2$, $G_2$ and $F_4$ in terms of the mod-2 Galois cohomological invariants attached with these groups. The results in this paper and our earlier paper ([6]) show that the mod-2 invariants of groups of type $G_2,F_4$ and $A_2$ are controlled by their k-subgroups of type $A_1$ and $A_2$ as well as the unitary k-tori embedded in them.     

Proper connection and size of graphs

An edge-coloured graph $G$ is called properly connected if any two vertices are connected by a path whose edges are properly coloured. The proper connection number of a connected graph $G,$ denoted by $\mathrm{pc}\left(G\right)$, is the smallest number of colours that are needed in order to make $G$ properly connected. Our main result is the following: Let $G$ be a connected graph of order $n$ and $k\ge 2$. If $\left|E\left(G\right)\right|\ge \left(\genfrac{}{}{0ex}{}{n?k?1}{2}\right)+k+2$, then $\mathrm{pc}\left(G\right)\le k$ except when $k=2$ and $G\in \{G_1,G_2\},$ where $G_1=K_1\vee (2K_1+K_2)$ and $G_2=K_1\vee (K_1+2K_2).$     

Extremal product-one free sequences in Dihedral and Dicyclic Groups

Let $G$ be a finite group, written multiplicatively. The Davenport constant of $G$ is the smallest positive integer $D\left(G\right)$ such that every sequence of $G$ with $D\left(G\right)$ elements has a non-empty subsequence with product $1$. Let $D_{2n}$ be the Dihedral Group of order $2n$ and $Q_{4n}$ be the Dicyclic Group of order $4n$. Zhuang and Gao (2005) showed that $D\left(D_{2n}\right)=n+1$ and Bass (2007) showed that $D\left(Q_{4n}\right)=2n+1$. In this paper, we give explicit characterizations of all sequences $S$ of $G$ such that $\left|S\right|=D\left(G\right)?1$ and $S$ is free of subsequences whose product is 1, where $G$ is equal to $D_{2n}$ or $Q_{4n}$ for some $n$.     

Strong vertex-magic and edge-magic labelings of 2-regular graphs of odd order using Kotzig completion

Let $G$ be a 2-regular graph with $2m+1$ vertices and assume that $G$ has a strong vertex-magic total labeling. It is shown that the four graphs $G\cup 2m{C}_3$, $G\cup (2m+2)C_3$, $G\cup m{C}_8$ and $G\cup (m+1)C_8$ also have a strong vertex-magic total labeling. These theorems follow from a new use of carefully prescribed Kotzig arrays. To illustrate the power of this technique, we show how just three of these arrays, combined with known labelings for smaller 2-regular graphs, immediately provide strong vertex-magic total labelings for 68 different 2-regular graphs of order 49.     

Ramsey numbers of odd cycles versus larger even wheels

The generalized Ramsey number $R(G_1,G_2)$ is the smallest positive integer $N$ such that any red–blue coloring of the edges of the complete graph $K_N$ either contains a red copy of $G_1$ or a blue copy of $G_2$. Let $C_m$ denote a cycle of length $m$ and $W_n$ denote a wheel with $n+1$ vertices. In 2014, Zhang, Zhang and Chen determined many of the Ramsey numbers $R(C_{2k+1},W_n)$ of odd cycles versus larger wheels, leaving open the particular case where $n=2j$ is even and $k<j<3k∕2$. They conjectured that for these values of $j$ and $k$, $R(C_{2k+1},W_{2j})=4j+1$. In 2015, Sanhueza-Matamala confirmed this conjecture asymptotically, showing that $R(C_{2k+1},W_{2j})\le 4j+334$. In this paper, we prove the conjecture of Zhang, Zhang and Chen for almost all of the remaining cases. In particular, we prove that $R(C_{2k+1},W_{2j})=4j+1$ if $j?k\ge 251$, $k<j<3k∕2$, and $j\ge 212299$.