edge distance-balanced graphs are graphs in which for every edge $e = uv$ the number of edges closer to vertex $u$ than to vertex $v$ is equal to the number of edges closer to $v$ than to $u$. in this paper, we study this property under some graph operations.

This paper describes an empirical investigation into the nature of challenge within the context of flow or effortless attention. An important condition of the flow state is that the challenges of the task are commensurate with the person’s level of skill. However when considering movement interaction those challenges could be composite in the sense that they could comprise physical as well as i...

The blocks of a balanced ternary design, BTD(V,B; ρ1, ρ2, R;K,Λ), can be partitioned into two sets: the b1 blocks that each contain no repeated elements, and the b2 = B − b1 blocks containing repeated elements. In this note, we address, and answer in some particular cases, the following question. For which partitions of the integer B as b1 + b2 does there exist a BTD(V,B; ρ1, ρ2, R;K,Λ)?

Given a set of n black and n white points in general position in the plane, a line l determined by them is said to be balanced if each open half-plane bounded by l contains precisely the same number of black points as white points. It is proved that the number of balanced lines is at least n. This settles a conjecture of George Baloglou.

Let S be a set of r red points and b = r + 2δ blue points in general position in the plane, with δ ≥ 0. A line l determined by them is balanced if in each open half-plane bounded by l the difference between the number of blue points and red points is δ. We show that every set S as above has at least r balanced lines. The main techniques in the proof are rotations and a generalization, sliding r...

A balanced pair in an ordered set P = (V,6) is a pair (x, y) of elements of V such that the proportion of linear extensions of P that put x before y is in the real interval [1/3, 2/3]. We prove that every finite N -free ordered set which is not totally ordered has a balanced pair.

Let B and R be point sets (of blue and red points, respectively) in the plane, such that P := B ∪R is in general position, and |P | is even. A line l is balanced if it spans one blue and one red point, and on each open halfplane of l, the number of blue points minus the number of red points is the same. We prove that P has at least min{|B|, |R|} balanced lines. This refines a result by Pach and...