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A problem asking for the shortest tour of a graph which visits each edge at least once (Kwan 1962; Skiena 1990, p. 194). For an Eulerian graph, an Eulerian cycle is the ...

A fundamental structural result in extremal graph theory due to Szemerédi (1978). The regularity lemma essentially says that every graph can be well-approximated by the union ...

The clique covering number theta(G) of a graph G is the minimum number of cliques in G needed to cover the vertex set of G. Since theta(G) involves the minimum number of ...

A two-coloring of a complete graph K_n of n nodes which contains exactly the number of monochromatic forced triangles and no more (i.e., a minimum of R+B where R and B are ...

Two nonisomorphic graphs can share the same graph spectrum, i.e., have the same eigenvalues of their adjacency matrices. Such graphs are called cospectral. For example, the ...

The transitive closure of a binary relation R on a set X is the minimal transitive relation R^' on X that contains R. Thus aR^'b for any elements a and b of X provided that ...

The so-called reaching algorithm can solve the shortest path problem (i.e., the problem of finding the graph geodesic between two given nodes) on an m-edge graph in O(m) ...

An s-route of a graph G is a sequence of vertices (v_0,v_1,...,v_s) of G such that v_iv_(i+1) in E(G) for i=0, 1, ..., s-1 (where E(G) is the edge set of G) and ...

A generalization of Ramsey theory to mathematical objects in which one would not normally expect structure to be found. For example, there exists a graph with very few ...

The fractional edge chromatic number of a graph G is the fractional analog of the edge chromatic number, denoted chi_f^'(G) by Scheinerman and Ullman (2011). It can be ...