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A transitive subgroup $G\le S_p$ for prime $p$ that contains a transposition is $S_p$. Form a graph on the $p$ letters with edges given by all conjugates in $G$ of that transposition. The transitive action permutes connected components, so their common size divides $p$. An edge rules out size one, hence the graph is connected. Its edge transpositions generate $S_p$: successive swaps along a path generate the swap of its endpoint with any chosen start. All these edge transpositions belong to $G$.

In a dense graph with many common neighbours for every pair of vertices, choose disjoint random small sets. A set is good if few vertices have no neighbour in it. The Markov inequality controls the number of bad sets, and random sampling makes almost every pair of good sets adjacent. Add unused common neighbours to connect each set and to repair the remaining missing adjacencies. If the number of used vertices stays below every common-neighbour count, this constructs disjoint branch sets of a graph minor representing a large complete graph.

A vertex $v$ is reachable from $u$ if a finite path leads from $u$ to $v$, respecting edge directions in a directed graph. For a finite graph, graph search decides reachability by maintaining the set of discovered vertices; every step discovers a new vertex or examines an edge, so the procedure terminates.

If a set partition refines a rectangle $U\times V$ of a graph into rectangles $U_a\times V_b$, write $d=d(U,V)$ for its edge density of a bipartite graph. The weighted variance identity isIt follows by expanding the square and using ${\sum }_{a,b}|U_a||V_b|d(U_a,V_b)=|U||V|d$. In particular, equitable regularity energy cannot decrease under refinement. If a witness rectangle has relative sizes at least $\epsilon$ and edge density of a bipartite graph discrepancy greater than $\epsilon$, its contribution, after division by the squared total vertex count, exceeds ${\epsilon }^4|U||V|/N^2$.

For a graph $\Gamma$, the right-angled Artin group is

In independent bond percolation on a graph, let $c_n$ be the number of $n$-step self-avoiding walks from a fixed graph vertex and ${\kappa }_n$ the number that are open. Then $\mathbb{E}{\kappa }_n=c_n{p}^n$. The Jensen inequality for the concave function $t^{1/n}$ gives $\mathbb{E}({\kappa }_n^{1/n})\le p{c}_n^{1/n}$. No independence between different self-avoiding walk indicators is needed. When $c_n^{1/n}\to \mu$, the limit superior is at most $p\mu$.

An $n$-step self-avoiding walk on a graph is a sequence $v_0,\dots ,v_n$ of distinct graph vertices with an edge between successive graph vertices. It is a graph path with $n$ edges and $n+1$ graph vertices. On a transitive graph, the exponential growth rate of rooted self-avoiding walk counts is the connective constant.

A $k$-semicoloring of a finite graph assigns one of $k$ colors to every vertex and is a proper graph colouring on an induced subgraph containing at least half the vertices. This is weaker than a proper coloring of the whole graph, and is useful as an intermediate step in a randomized algorithm.

Suppose a graph with $n$ vertices and $m>0$ edges has a vector coloring with adjacent inner products at most $-1/2$. Take $r$ independent copies of random hyperplane rounding and use the $r$ signs as a color. Each edge remains monochromatic with probability at most $3^{-r}$, so the expected number $B$ of monochromatic edges is at most $m{3}^{-r}$ by linearity of expectation.

Choose $r=\max \{0,\lceil {\log }_3(4m/n)\rceil \}$. Then Markov inequality gives $\mathbb{P}(B>n/2)\le 1/2$. On success, delete one endpoint of every monochromatic edge. This random alteration method leaves at least $n/2$ vertices properly colored withThe removed vertices can retain their original colors, as the semicoloring only requires the retained induced subgraph to be proper. A failed draw can be detected and repeated, with at most two trials on average. The case $m=0$ needs just one color. This is the elementary independent-hyperplane construction in Karger, Motwani and Sudan's paper on approximate graph coloring.

Offer $k$ independent uniform edges of the complete graph per row and permit any one to be selected from each row. If a chosen graph has no graph component of order at least $2n/3$, the balanced component cut provides an empty cut whose sides both have at least $n/3$ vertices. Each candidate edge crosses that cut with probability at least $4/9$, so a row can avoid crossing with probability at most $1-(4/9{)}^k$. The union bound over at most $2^n$ cuts gives failure probability at most $2^n[1-(4/9{)}^k{]}^{cn}$. Any fixed integer $c$ with $c[-\log (1-(4/9{)}^k)]>\log 2$ suffices. The guarantee is simultaneous even for choices made after seeing every offered edge.

The speed counts the labelled graphs with a given property on vertex set $[n]$. For proper unbounded hereditary graph properties, its base-two logarithm has leading term $(1-1/r)\left(\genfrac{}{}{0ex}{}{n}{2}\right)$, where $r$ is the property's colouring number.

In a cycle square, each triple of consecutive vertices is a clique. A proper three-colouring therefore repeats with period three and closes exactly when the length is divisible by three. For all other cycle lengths at least four colours are needed; the five-cycle square is a complete five-vertex graph.

A strict vector $k$-coloring requires equality $⟨v_i,v_j⟩=-1/(k-1)$ on every edge. The least admissible $k$ is the complement theta number for a graph with an edge. Allowing merely an inequality defines the potentially smaller vector chromatic number.

Membership is preserved under deleting vertices and edges. This is the decreasing meaning of monotonicity in extremal graph enumeration; it differs from an increasing monotone graph property used for random-graph thresholds. For a proper unbounded class, its least excluded chromatic number determines the leading quadratic logarithm of its labelled speed.

A tree component is a graph component whose induced graph is a tree. A tree component with $j$ vertices has exactly $j-1$ edges.

The triangle count of a finite simple graph is the number of unordered triples of vertices inducing a triangle in a graph. In a tripartite graph with parts $X,Y,Z$ and adjacency indicator functions $g_{XY},g_{YZ},g_{XZ}$, it equals $|X||Y||Z|{\mathbb{E}}_{x,y,z}{g}_{XY}(x,y)g_{YZ}(y,z)g_{XZ}(x,z)$, using uniform expectations on nonempty parts.

For every $\eta >0$, some $\rho >0$ and $N_0$ have this property: a graph on $N\ge N_0$ vertices with fewer than $\rho N^3$ unordered triangles in a graph can be made triangle-free by deleting at most $\eta N^2$ edges. A Szemerédi regularity lemma partition deletes exceptional, intraclass, irregular and sparse-pair edges cheaply. A surviving triangle in a graph would force a positive cubic count by the regular triangle counting lemma.

For $n\ge 3$, put $e_i=e(T_i(n))$, $t_3=k_3(T_3(n))$, and $c_n=(e_3-e_2)/t_3$. Every graph on $n$ vertices satisfiesThus an edge count $e_2+\theta (e_3-e_2)$ forces at least $\theta t_3$ triangles in a graph. This is an exact finite bound, with the rounding in the Turan graph retained. To prove it, use edge-triangle symmetrization. The inequality $c_n\ge 2/n$ allows merging the two smallest classes when there are at least four classes. With three classes, fixing the middle class makes the objective affine in the product of the other two sizes; either merge them or balance them. Only $T_2(n)$ and $T_3(n)$ need remain.