Arc-Transitive Graphs of Square-free Order with Valency 11

In this paper, we present a complete list of connected arc-transitive graphs of square-free order with valency 11. The list includes the complete bipartite graph [Formula: see text], the normal Cayley graphs of dihedral groups and the graphs associated with the simple group [Formula: see text] and [Formula: see text], where [Formula: see text] is a prime.

The Maximum Number of Copies of $K_{r,s}$ in Graphs Without Long Cycles or Paths

The circumference of a graph is the length of a longest cycle of it. We determine the maximum number of copies of $K_{r,s}$, the complete bipartite graph with classes sizes $r$ and $s$, in 2-connected graphs with circumference less than $k$. As corollaries of our main result, we determine the maximum number of copies of $K_{r,s}$ in $n$-vertex $P_{k}$-free and $M_k$-free graphs for all values of $n$, where $P_k$ is a path on $k$ vertices and $M_k$ is a matching on $k$ edges.

Martingales and the characteristic functions of absorption time on bipartite graphs

Evolutionary graph theory investigates how spatial constraints affect processes that model evolutionary selection, e.g. the Moran process. Its principal goals are to find the fixation probability and the conditional distributions of fixation time, and show how they are affected by different graphs that impose spatial constraints. Fixation probabilities have generated significant attention, but much less is known about the conditional time distributions, even for simple graphs. Those conditional time distributions are difficult to calculate, so we consider a close proxy to it: the number of times the mutant population size changes before absorption. We employ martingales to obtain the conditional characteristic functions (CCFs) of that proxy for the Moran process on the complete bipartite graph. We consider the Moran process on the complete bipartite graph as an absorbing random walk in two dimensions. We then extend Wald’s martingale approach to sequential analysis from one dimension to two. Our expressions for the CCFs are novel, compact, exact, and their parameter dependence is explicit. We show that our CCFs closely approximate those of absorption time. Martingales provide an elegant framework to solve principal problems of evolutionary graph theory. It should be possible to extend our analysis to more complex graphs than we show here.

On Finite Subnormal Cayley Graphs

In this paper we introduce and study a type of Cayley graph – subnormal Cayley graph. We prove that a subnormal 2-arc transitive Cayley graph is a normal Cayley graph or a normal cover of a complete bipartite graph $\mathbf{K}_{p^d,p^d}$ with $p$ prime. Then we obtain a generic method for constructing half-symmetric (namely edge transitive but not arc transitive) Cayley graphs.

LIST COLORING OF BLOCK GRAPHS AND COMPLETE BIPARTITE GRAPHS

List coloring is a vertex coloring of a graph where each vertex can be restricted to a list of allowed colors. For a given graph G and a set L(v) of colors for every vertex v, a list coloring is a function that maps every vertex v to a color in the list L(v) such that no two adjacent vertices receive the same color. It was first studied in the 1970s in independent papers by Vizing and by Erdős, Rubin, and Taylor. A block graph is a type of undirected graph in which every biconnected component (block) is a clique. A complete bipartite graph is a bipartite graph with partitions V 1, V 2 such that for every two vertices v_1∈V_1 and v_2∈V_2 there is an edge (v 1, v 2). If |V_1 |=n and |V_2 |=m it is denoted by K_(n,m). In this paper we provide a polynomial algorithm for finding a list coloring of block graphs and prove that the problem of finding a list coloring of K_(n,m) is NP-complete even if for each vertex v the length of the list is not greater than 3 (|L(v)|≤3).

Note on Path-Connectivity of Complete Bipartite Graphs

For a graph [Formula: see text] and a set [Formula: see text] of size at least [Formula: see text], a path in [Formula: see text] is said to be an [Formula: see text]-path if it connects all vertices of [Formula: see text]. Two [Formula: see text]-paths [Formula: see text] and [Formula: see text] are said to be internally disjoint if [Formula: see text] and [Formula: see text]. Let [Formula: see text] denote the maximum number of internally disjoint [Formula: see text]-paths in [Formula: see text]. The [Formula: see text]-path-connectivity [Formula: see text] of [Formula: see text] is then defined as the minimum [Formula: see text], where [Formula: see text] ranges over all [Formula: see text]-subsets of [Formula: see text]. In [M. Hager, Path-connectivity in graphs, Discrete Math. 59 (1986) 53–59], the [Formula: see text]-path-connectivity of the complete bipartite graph [Formula: see text] was calculated, where [Formula: see text]. But, from his proof, only the case that [Formula: see text] was considered. In this paper, we calculate the situation that [Formula: see text] and complete the result.