We consider the following generalization of graph packing. Let G 1 = (V 1 , E 1 ) and G 2 = (V 2 , E 2 ) be graphs of order n and GWe extend the classical results of Sauer and Spencer and Bollobás and Eldridge on packing of graphs with small sizes or maximum degrees to the setting of list packing. In particular, we extend the well-known Bollobás-Eldridge Theorem, proving that if ∆(G 1 ) ≤ n−2, ∆(G 2 ) ≤ n−2, ∆(G 3 ) ≤ n−1, and |E 1 |+|E 2 |+|E 3 | ≤ 2n−3, then either (G 1 , G 2 , G 3 ) packs or is one of 7 possible exceptions. Hopefully, the concept of list packing will help to solve some problems on ordinary graph packing, as the concept of list coloring did for ordinary coloring.Mathematics Subject Classification: 05C70, 05C35.

Given graphs $G$ and $H$, $G$ is $H$-saturated if $H$ is not a subgraph of $G$, but for all $e \notin E(G)$, $H$ appears as a subgraph of $G + e$. While for every $n \ge |V(H)|$, there exists an $n$-vertex graph that is $H$-saturated, the same does not hold for induced subgraphs. That is, there exist graphs $H$ and values of $n \ge |V(H)|$, for which every $n$-vertex graph $G$ either contains $H$ as an induced subgraph, or there exists $e \notin E(G)$ such that $G + e$ does not contain $H$ as an induced subgraph. To circumvent this Martin and Smith make use of a generalized notion of "graph" when introducing the concept of induced saturation and the induced saturation number of graphs. This allows for edges that can be included or excluded when searching for an induced copy of $H$, and the induced saturation number is the minimum number of such edges that are required.In this paper, we show that the induced saturation number of many common graphs is zero. This yields graphs that are $H$-induced-saturated. That is, graphs such that no induced copy of $H$ exists, but adding or deleting any edge creates an induced copy of $H$. We introduce a new parameter for such graphs, indsat*($n;H$), which is the minimum number of edges in an $H$-induced-saturated graph. We provide bounds on indsat*($n;H$) for many graphs. In particular, we determine indsat*($n;H$) completely when $H$ is the paw graph $K_{1,3}+e$, and we determine indsat*(n;$K_{1,3}$) within an additive constant of four.

This work presents a new method to quantify connectivity in transportation networks. Inspired by the field of topological data analysis, we propose a novel approach to explore the robustness of road network connectivity in the presence of congestion on the roadway. The robustness of the pattern is summarized in a congestion barcode, which can be constructed directly from traffic datasets commonly used for navigation. As an initial demonstration, we illustrate the main technique on a publicly available traffic dataset in a neighborhood in New York City. arXiv:1707.08557v1 [physics.soc-ph]

Let G = (V, E) be a graph on n vertices and f : V → [1, n] a one to one map of V onto the integers 1 through n.Define the bandwidth B(G) of G to be the minimum possible value of dilation(f ) over all such one to one maps f . Next define the Kneser Graph K(n, r) to be the graph with vertex set [n] r , the collection of r-subsets of an n element set, and edge set E = {vw : v, w ∈[n] r , v ∩ w = ∅}. For fixed r ≥ 4 and n → ∞ we show that *

Dedicated to Adrian Bondy on the occasion of his 70 th birthday.then G 1 and G 2 pack. In the same paper, he conjectured that if ∆(G 1 ), ∆(G 2 ) ≤ n − 2, then |E 1 | + |E 2 | + max{∆(G 1 ), ∆(G 2 )} ≤ 3n − 7 is sufficient for G 1 and G 2 to pack. We prove that, up to an additive constant,Żak's conjecture is correct. Namely, there is a constant C such that if ∆(G 1 ), ∆(G 2 ) ≤ n − 2 and |E 1 |+|E 2 |+max{∆(G 1 ), ∆(G 2 )} ≤ 3n−C, then G 1 and G 2 pack. In order to facilitate induction, we prove a stronger result on list packing.