On circle graphs with girth at least five

Electronic Notes in Discrete Mathematics 29 (2007) 129–133 www.elsevier.com/locate/endm

1

g≥5

g−4

g = 5

g≥5

C C − −→ x− i xi+1 0 ≤ i ≤ k−1

{x0 , . . . , xk−1 } ⊂ C

i

C (x1 x2 y1 y2 )

k y

x

(x0 , . . . , xk−1 ) −−−→ x i+1 xi 0 ≤ i ≤ k−1 {x, y} C {x1 , y1 } {x2 , y2}

G ≤

k k

k

≥

k

k

1

1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2007.07.022

G k

130

L. Esperet, P. Ochem / Electronic Notes in Discrete Mathematics 29 (2007) 129–133

C

G V (G) = {v1 , . . . , vn } C = {{x1 , y1 }, . . . , {xn , yn }} i, j vi G {xi , yi} {xj , yj }

vj

G

C

G

c x G  χc (G) =

C

≤

g≥5

(g − 4)

G

C C

G y

1

1≤q ≤p

(p, q) {0, . . . , p − 1} q ≤ |c(x) − c(y)| ≤ p − q 

p | q

(p, q)

χ(G) − 1 < χc (G) ≤ χ(G)

χ(G) = χc (G)

(g −4) G

g≥5

1 χc (G) ≤ 2 +  g−3  . 2

G .

C

131

L. Esperet, P. Ochem / Electronic Notes in Discrete Mathematics 29 (2007) 129–133

G (G) =

{

(H), H ⊆ G}

(H) =

G 2g/(g − 2)

g

g

(Gn ) → 2g/(g − 2)

G

Seg

1-String

Planar

μg

2+

4 g−2

(G) | G ∈ F

(Gn )n≥0 n → ∞

(G) <

G

g} .

μg

g

Outerplanar

2+

(G) <

g ≥5

2 + 2/(g − 4).

μg (F ) = sup {

2|E(H)| . |V (H)|

μg

Partial 2-Tree

2 g−2

2+

2

 g−1 2

Seg

2+

4 g−4

1-String

2+

4 g−4

μg

μg

g ≥ 5 μg (Circle) = 2

G = (V, E) C = {{x1 , x1 }, . . . , {xn , xn }} C



g

g−2 g−4

g≥5

G

132

L. Esperet, P. Ochem / Electronic Notes in Discrete Mathematics 29 (2007) 129–133

{x, x } {y, y } (xyzz  y x )

{y, y } {y, y } Cr

Cb

Gr = G(Cr )

{z, z  }

Gb = G(Cb )

Cr − → yx {u, v} A2

− → xy

→ > 0} A1 ∪ A2 , ρ(− xy) t

− → xy

{x, y} A1 {x, y}

− → − → − → ux xv vy

− → yu

t=

→ − →∈ {ρ(− xy), xy

→ ρ(− xy)

− → xy

t g−4

g

Qg,0 = Cg Qg,t

g=5

 μg (Circle) ≥ lim

t→∞

Q5,0

g

(Qg,t ) = 2

Q5,1

g ≥ 5 Qg,t+1

g−2 g−4

(Qg,t )t≥0

133

L. Esperet, P. Ochem / Electronic Notes in Discrete Mathematics 29 (2007) 129–133

2



G g−2 g−4

g ≥ 5

(G) ≤

G

G

C

R(C) R(C)

•

C R(C) C

•

R(C) G

ω Ω(ω

ω)

2

ω+6