A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the late 1960s. In the intervening years dozens of graph labelings techniques have been studied in over 1000 papers. Finding out what has been done for any particular kind of labeling and keeping up with new discoveries is difficult because of the sheer number of papers and because many of the papers have appeared in journals that are not widely available. In this survey I have collected everything I could find on graph labeling. For the convenience of the reader the survey includes a detailed table of contents and index.

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A Dynamic Survey of Graph Labeling

The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. General upper bounds - called Moore bounds - for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bounds for the maximum possible number of vertices, given the other two parameters, and thus attacking the degree/diameter problem 'from above', remains a largely unexplored area. Constructions producing large graphs and digraphs of given degree and diameter represent a way of attacking the degree/diameter problem 'from below'. This survey aims to give an overview of the current state-of-the-art of the degree/diameter problem. We focus mainly on the above two streams of research. However, we could not resist mentioning also results on various related problems. These include considering Moore-like bounds for special types of graphs and digraphs, such as vertex-transitive, Cayley, planar, bipartite, and many others, on the one hand, and related properties such as connectivity, regularity, and surface embeddability, on the other hand.

Moore Graphs and Beyond: A survey of the Degree/Diameter Problem

On a Problem in Combinations

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.

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A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas

In many applications of graph algorithms, including communication networks, graphics, assembly
planning, and VLSI design, graphs are subject to discrete changes, such as additions or deletions of
edges or vertices. In the last decades there has been a growing interest in such dynamically changing
graphs, and a whole body of algorithms and data structures for dynamic graphs has been discovered.
This chapter is intended as an overview of this ﬁeld.
In a typical dynamicgraphproblemonewould like to answerqueries ongraphs that areundergoinga sequence of updates, for instance, insertions and deletions of edges and vertices. The goal of a
dynamic graph algorithm is to update eﬃciently the solution of a problem after dynamic changes,
rather than having to recompute it from scratch each time. Given their powerful versatility, it is not
surprising that dynamic algorithms and dynamic data structures are often more diﬃcult to design
and analyze than their static counterparts.
We can classify dynamic graph problems according to the types of updates allowed. A problemis said to be fully dynamic if the update operations include unrestricted insertions and deletions of
edges. A problem is called partially dynamic if only one type of update, either insertions or deletions,
is allowed. If only insertions are allowed, the problem is called incremental; if only deletions are
allowed it is called decremental.
In this chapter, we describe fully dynamic algorithms for graph problems. We present dynamicalgorithms for undirected graphs in Section 9.2. Dynamic algorithms for directed graphs are next
described in Section 9.3. Finally, in Section 9.4 we describe some open problems.

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Dynamic graph algorithms

A Note on Large Graphs of Diameter Two and Given Maximum Degree

We present a theory of Cayley maps, i.e., embeddings of Cayley graphs into oriented surfaces having the same cyclic rotation of generators around each vertex. These maps have often been used to encode symmetric embeddings of graphs. We also present an algebraic theory of Cayley maps and we apply the theory to determine exactly which regular or edge-transitive tilings of the sphere or plane are Cayley maps or Cayley graphs. Our main goal, however, is to provide the general theory so as to make it easier for others to study Cayley maps.

Cayley maps

http://mcs.open.ac.uk/mjg47/papers/nonisoem.pdf

Exponential Families of Non-Isomorphic Triangulations of Complete Graphs

An enumeration result for orientably regular hypermaps of a given type with automorphism groups isomorphic to PSL(2,q) or PGL(2,q) can be extracted from a 1969 paper by Sah. We extend the investigation to orientable reflexible hypermaps and to nonorientable regular hypermaps, providing many more details about the associated computations and explicit generating sets for the associated groups.

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Regular hypermaps over projective linear groups

This paper uses combinatorial group theory to help answer some long-standing questions about the genera of orientable surfaces that carry particular kinds of regular maps By classifying all orientably-regular maps whose automorphism group has order coprime to g −1, where g is the genus, all orientably-regular maps of genus p+1 for p prime are determined As a consequence, it is shown that orientable surfaces of inﬁnitely many genera carry no regular map that is chiral (irreﬂexible), and that orientable surfaces of inﬁnitely many genera carry no reﬂexible regular map with simple underlying graph Another consequence is a simpler proof of the Breda–Nedela–Siraň classiﬁcation of non-orientable regular maps of Euler characteristic −p where p is prime

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Jozef Širáň

Papers

A Note on Large Graphs of Diameter Two and Given Maximum Degree

Cayley maps

Exponential Families of Non-Isomorphic Triangulations of Complete Graphs

Regular hypermaps over projective linear groups

The genera, reflexibility and simplicity of regular maps