Extremal Numbers of Cycles Revisited

Extremal graphs without 4-cycles

We prove an upper bound for the number of edges a C4-free graph on q 2 + q vertices can contain for q even. This upper bound is achieved whenever there is an orthogonal polarity graph of a plane of even order q. Let n be a positive integer and G a graph. We define ex(n,G) to be the largest number of edges possible in a graph on n vertices that does not contain G as a subgraph; we call a graph o...

Extremal Permutations in Routing Cycles

Let G be a graph whose vertices are labeled 1, . . . , n, and π be a permutation on [n] := {1, 2, . . . , n}. A pebble pi that is initially placed at the vertex i has destination π(i) for each i ∈ [n]. At each step, we choose a matching and swap the two pebbles on each of the edges. Let rt(G, π), the routing number for π, be the minimum number of steps necessary for the pebbles to reach their d...

Extremal graphs without three-cycles or four-cycles

We derive bounds for f(v), the maximum number of edges in a graph on v vertices that contains neither three-cycles nor four-cycles. Also, we give the exact value of f(v) for all v up to 24 and constructive lower bounds for all v up to 200.

Moore Graphs and Cycles Are Extremal Graphs for Convex Cycles

Let ρ(G) denote the number of convex cycles of a simple graph G of order n, size m, and girth 3 ≤ g ≤ n. It is proved that ρ(G) ≤ ng (m−n+ 1) and that equality holds if and only if G is an even cycle or a Moore graph. The equality also holds for a possible Moore graph of diameter 2 and degree 57 thus giving a new characterization of Moore graphs.

Trees with extremal numbers of dominating sets