We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

We survey recent advances in algorithms for route planning in transportation networks. For road networks, we show that one can compute driving directions in milliseconds or less even at continental scale. A variety of techniques provide different trade-offs between preprocessing effort, space requirements, and query time. Some algorithms can answer queries in a fraction of a microsecond, while others can deal efficiently with real-time traffic. Journey planning on public transportation systems, although conceptually similar, is a significantly harder problem due to its inherent time-dependent and multicriteria nature. Although exact algorithms are fast enough for interactive queries on metropolitan transit systems, dealing with continent-sized instances requires simplifications or heavy preprocessing. The multimodal route planning problem, which seeks journeys combining schedule-based transportation (buses, trains) with unrestricted modes (walking, driving), is even harder, relying on approximate solutions even for metropolitan inputs.

/pdf/route-planning-in-transportation-networks-k0typz8i4k.pdf

Route Planning in Transportation Networks

Abraham et al. [SODA 2010] have recently presented a theoretical analysis of several practical point-to-point shortest path algorithms based on modeling road networks as graphs with low highway dimension. They also analyze a labeling algorithm. While no practical implementation of this algorithm existed, it has the best time bounds. This paper describes an implementation of the labeling algorithm that is faster than any existing method on continental road networks.

/pdf/a-hub-based-labeling-algorithm-for-shortest-paths-in-road-2xtonimcev.pdf

A hub-based labeling algorithm for shortest paths in road networks

A signed graph is a graph whose edges are labeled by signs. This is a bibliography of signed graphs and related mathematics. Several kinds of labelled graph have been called "signed" yet are mathematically very different. I distinguish four types: Group-signed graphs: the edge labels are elements of a 2-element group and are multiplied around a polygon (or along any walk). Among the natural generalizations are larger groups and vertex signs. Sign-colored graphs, in which the edges are labelled from a two-element set that is acted upon by the sign group: - interchanges labels, + leaves them unchanged. This is the kind of "signed graph" found in knot theory. The natural generalization is to more colors and more general groups —  or no group. Weighted graphs, in which the edge labels are the elements +1 and -1 of the integers or another additive domain. Weights behave like numbers, not signs; thus I regard work on weighted graphs as outside the scope of the bibliography — ex cept (to some extent) when the author calls the weights "signs". Labelled graphs where the labels have no structure or properties but are called "signs" for any or no reason.

/pdf/a-mathematical-bibliography-of-signed-and-gain-graphs-and-3a49f10hs2.pdf

A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas

In this paper we analyze the complexity of algorithms for two problems that arise in automatic test path generation for programs: the problem of building a path through a specified set of program statements and the problem of building a path which satisfies impossible-pairs restrictions on statement pairs. These problems are both reduced to graph traversal problems. We give an efficient algorithm for the first, and show that the second is NP-complete.

On Two Problems in the Generation of Program Test Paths ; CU-CS-081-75

For a given number L, an L-length-bounded edge-cut (node-cut, respectively) in a graph G with source s and sink t is a set C of edges (nodes, respectively) such that no s-t-path of length at most L remains in the graph after removing the edges (nodes, respectively) in C. An L-length-bounded flow is a flow that can be decomposed into flow paths of length at most L. In contrast to classical flow theory, we describe instances for which the minimum L-length-bounded edge-cut (node-cut, respectively) is Θ(n2/3)-times (Θ(√n)-times, respectively) larger than the maximum L-length-bounded flow, where n denotes the number of nodes; this is the worst case. We show that the minimum length-bounded cut problem is NP-hard to approximate within a factor of 1.1377 for L≥ 5 in the case of node-cuts and for L≥ 4 in the case of edge-cuts. We also describe algorithms with approximation ratio O(minlL,n/Lr) ⊆ O√n in the node case and O(min lL,n2/L2,√mr ⊆ O2/3 in the edge case, where m denotes the number of edges. Concerning L-length-bounded flows, we show that in graphs with unit-capacities and general edge lengths it is NP-complete to decide whether there is a fractional length-bounded flow of a given value. We analyze the structure of optimal solutions and present further complexity results.

/pdf/length-bounded-cuts-and-flows-59scplzqpx.pdf

Length-bounded cuts and flows

http://kam.mff.cuni.cz/~kolman/papers/fpath3.pdf

On the complexity of paths avoiding forbidden pairs

We study a concept of group coloring introduced by Jaeger et al. We show that the group chromatic number of a graph with minimum degree δ is greater than δ-(2, ln δ) and we answer several open questions on the group chromatic number of planar graphs: a construction of a bipartite planar graph with group chromatic number four and a 3-colorable planar graph with group chromatic number five are presented. We also observe that several upper bounds on the choice number for various subclasses of planar graphs also translate to the concept of group colorings. © 2005 Wiley Periodicals, Inc. J Graph Theory

A note on group colorings

We develop an idea of a local 3-edge-coloring of a cubic graph, a generalization of the usual 3-edge-coloring. We allow for an unlimited number of colors but require that the colors of two edges meeting at a vertex always determine the same third color. Local 3-edge-colorings are described in terms of colorings by points of a partial Steiner triple system such that the colors meeting at each vertex form a triple of the system. An important place in our investigation is held by the two smallest non-trivial Steiner triple systems, the Fano plane PG(2,2) and the affine plane AG(2,3). For i=4,5, and 6 we identify certain configurations F"i and A"i of i lines of the Fano plane and the affine plane, respectively, and prove a theorem saying that a cubic graph admits an F"i-coloring if and only if it admits an A"i-coloring. Among consequences of this is the result of Holroyd and Skoviera [F. Holroyd, M. Skoviera, Colouring of cubic graphs by Steiner triple systems, J. Combin. Theory Ser. B 91 (2004) 57-66] that the edges of every bridgeless cubic graph can be colored by using points and blocks of any non-trivial Steiner triple system S. Another consequence is that every bridgeless cubic graph has a proper edge-coloring by elements of any abelian group of order at least 12 such that around each vertex the group elements sum to 0. We also propose several conjectures concerning edge-coloring of cubic graphs and relate them to several well-known conjectures. In particular, we show that both the Cycle Double Cover Conjecture and the Fulkerson Conjecture can be formulated as a coloring problem in terms of known geometric configurations - the Desargues configuration and the Cremona-Richmond configuration, respectively.

Ondřej Pangrác

Papers

Length-bounded cuts and flows

On the complexity of paths avoiding forbidden pairs

A note on group colorings

Projective, affine, and abelian colorings of cubic graphs

Notes: Markov bases of binary graph models of K4-minor free graphs