in the lex and colex orderings on $E(K_n)$. The first few edges in the lex ordering on $E(K_n)$ are $$\set{1,2},\set{1,3},\dots,\set{1,n},\set{2,3},\set{2,4},\dots,\set{2,n},\set{3,4},\dots$$ and the first few edges in the colex ordering on $E(K_n)$ are $$\set{1,2},\set{1,3},\set{2,3},\set{1,4},\set{2,4},\set{3,4},\set{1,5},\set{2,5},\dots$$

The *lex graph* ${\mathcal{L}}(n,e)$ is the graph with vertex set $[n]$ and edge set consisting of the first $e$ edges in the lex order on $E(K_n)$. Similarly, the *colex graph* ${\mathcal{C}}(n,e)$ is the graph with vertex set $[n]$ and edge set consisting of the first $e$ edges in the colex order on $E(K_n)$.

The graph below is ${\mathcal{L}}(7,8)$.

\(0) at (0,0) ; (1) at (1,0) ; (2) at (2,0) ; (3) at (3,0) ; (4) at (4,0) ; (5) at (5,0) ; (6) at (6,0) ;

\(0) circle (3pt) ; (1) circle (3pt) ; (2) circle (3pt) ; (3) circle (3pt) ; (4) circle (3pt) ; (5) circle (3pt) ; (6) circle (3pt) ;

=\[bend left\]

(0)(1)

(0)(2)

(0)(3)

(0)(4) (0)(5) (0)(6)

(1)(2)

(1)(3)

The graph below is ${\mathcal{C}}(7,8)$.

\(0) at (0,0) ; (1) at (1,0) ; (2) at (2,0) ; (3) at (3,0) ; (4) at (4,0) ; (5) at (5,0) ; (6) at (6,0) ;

\(0) circle (3pt) ; (1) circle (3pt) ; (2) circle 