For a given number of colors, $s$, the guessing graph is (base $s$) logarithm cardinality largest family colorings vertex set such that color each can be determined from colors vertices in its neighborhood. This quantity related to problems network coding, circuit complexity and entropy. We study graphs as property context classic extremal questions, relationship forbidden subgraph property. fi...

3 Third Lecture 11 3.1 Applications of the Zarankiewicz Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 The Turán Problem for Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 The Girth Problem and Moore’s Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4 Application of Moore’s Bound to Graph Spanners . . . . . . . . . . . ....

We survey various aspects of infinite extremal graph theory and prove several new results. The lead role play the parameters connectivity and degree. This includes the end degree. Many open problems are suggested.

Let χ(G) and χf (G) denote the chromatic and fractional chromatic numbers of a graph G, and let (n+, n0, n−) denote the inertia of G. We prove that: 1 + max ( n+ n− , n− n+ ) 6 χ(G) and conjecture that 1 + max ( n+ n− , n− n+ ) 6 χf (G). We investigate extremal graphs for these bounds and demonstrate that this inertial bound is not a lower bound for the vector chromatic number. We conclude with...

There are seven binary extremal self-dual doubly-even codes which are known to have a 2-transitive automorphism group. Using representation theoretical methods we show that there are no other such codes, except possibly n = 1 024. We also classify all extremal ternary self-dual and quaternary Hermitian self-dual codes.

Extremal graph theory is, broadly speaking, the study of relations between various graph invariants, such as order, size, connectivity, minimum/maximum degree, chromatic number, etc., and the values of these invariants that ensure that the graph has certain properties. Since the first major result by Turan in 1941, numerous mathematicians have contributed to make this a vibrant and deep subject...

We analyze the duration of the unbiased Avoider-Enforcer game for three basic positional games. All the games are played on the edges of the complete graph on n vertices, and Avoider’s goal is to keep his graph outerplanar, diamond-free and k-degenerate, respectively. It is clear that all three games are Enforcer’s wins, and our main interest lies in determining the largest number of moves Avoi...