Large butterfly Cayley graphs and digraphs

We present families of large undirected and directed Cayley graphs whose construction is related to butterfly networks. One approach yields, for every large $k$ and for values of $d$ taken from a large interval, the largest known Cayley graphs and digraphs of diameter $k$ and degree $d$. Another method yields, for sufficiently large $k$ and infinitely many values of $d$, Cayley graphs and digraphs of diameter $k$ and degree $d$ whose order is exponentially larger in $k$ than any previously constructed. In the directed case, these are within a linear factor in $k$ of the Moore bound.     

The relationship between k-forcing and k-power domination

Zero forcing and power domination are iterative processes on graphs where an initial set of vertices are observed, and additional vertices become observed based on some rules. In both cases, the goal is to eventually observe the entire graph using the fewest number of initial vertices. The concept of $k$-power domination was introduced by Chang et al. (2012) as a generalization of power domination and standard graph domination. Independently, $k$-forcing was defined by Amos et al. (2015) to generalize zero forcing. In this paper, we combine the study of $k$-forcing and $k$-power domination, providing a new approach to analyze both processes. We give a relationship between the $k$-forcing and the $k$-power domination numbers of a graph that bounds one in terms of the other. We also obtain results using the contraction of subgraphs that allow the parallel computation of $k$-forcing and $k$-power dominating sets.     

The perimeter of words

We define $\left[k\right]=\{1,2,\dots ,k\}$ to be a (totally ordered) alphabet on $k$ letters. A word $w$ of length $n$ on the alphabet $\left[k\right]$ is an element of ${\left[k\right]}^n$. A word can be represented by a bargraph (i.e., by a column-convex polyomino whose lower edges lie on the $x$-axis) in which the height of the $i$th column equals the size of the $i$th part of the word. Thus these bargraphs have heights which are less than or equal to $k$. We consider the perimeter, which is the number of edges on the boundary of the bargraph. By way of Cramer’s method and the kernel method, we obtain the generating function that counts the perimeter of words. Using these generating functions we find the average perimeter of words of length $n$ over the alphabet $\left[k\right]$. We also show how the mean and variance can be obtained using a direct counting method.     

Proper colouring Painter–Builder game

We consider the following two-player game, parametrised by positive integers $n$ and $k$. The game is played between Painter and Builder, alternately taking turns, with Painter moving first. The game starts with the empty graph on $n$ vertices. In each round Painter colours a vertex of her choice by one of the $k$ colours and Builder adds an edge between two previously unconnected vertices. Both players must adhere to the restriction that the game graph is properly $k$-coloured. The game ends if either all $n$ vertices have been coloured, or Painter has no legal move. In the former case, Painter wins the game; in the latter one, Builder is the winner. We prove that the minimal number of colours $k=k\left(n\right)$ allowing Painter’s win is of logarithmic order in the number of vertices $n$. Biased versions of the game are also considered.     

Maximum matchings in regular graphs

It was conjectured by Mkrtchyan, Petrosyan and Vardanyan that every graph $G$ with $\Delta \left(G\right)?\delta \left(G\right)\le 1$ has a maximum matching $M$ such that any two $M$-unsaturated vertices do not share a neighbor. The results obtained in Mkrtchyan et al. (2010), Petrosyan (2014) and Picouleau (2010) leave the conjecture unknown only for $k$-regular graphs with $4\le k\le 6$. All counterexamples for $k$-regular graphs $(k\ge 7)$ given in Petrosyan (2014) have multiple edges. In this paper, we confirm the conjecture for all $k$-regular simple graphs and also $k$-regular multigraphs with $k\le 4$.     

On degree sum conditions for 2-factors with a prescribed number of cycles

For a vertex subset $X$ of a graph $G$, let ${\Delta }_t\left(X\right)$ be the maximum value of the degree sums of the subsets of $X$ of size $t$. In this paper, we prove the following result: Let $k,m$ be positive integers, and let $G$ be an $m$-connected graph of order $n\ge 5k?2$. If ${\Delta }_2\left(X\right)\ge n$ for every independent set $X$ of size $?m∕k?+1$ in $G$, then $G$ has a 2-factor with exactly $k$ cycles. This is a common extension of the results obtained by Brandt et al. (1997) and Yamashita (2008), respectively.