? ? caterpil-

lar ? ? ? ? ? ? ? ? ? ? ? ? k i i ˇ Sáň

Šáň ? P_{n}

(5/3)^{n} ? ? ? ? ? lobster ? ? ? ? ? ü ? firecracker banana tree v v ? ? ? ? ? ? ? ? ? ? ? ? ? ? n C_{n} n ≡ 0 ? C_{n} n ≡ 1 W_{n} = C_{n}+ K_{1} ? ? ? n npoint suspension C_{m} C_{m}+ \overline{K_{n}} m ≡ 0 ? n m C_{m}+ \overline{K_{n}} ? C_{4}+ \overline{K_{n}}, C_{5}+ \overline{K_{2}}, C_{7}+ \overline{K_{n}}, C_{9}+ \overline{K_{2}}, C_{11}+

(\overline{K_{n}} C_{19}+) / (\overline{K_{n}} helm H_{n} n ? ? ? ? ? ? ? ? ? ? web ?) W (2, n)

W (t, n)

tn W (3, n)

W (4, n)

? ? W (5, n)

? W (t, 5)

5 ≤ t ≤ 13 ? ? closed helm flower gear graph

? ? n W_{n}

? ? C_{4} 2n C_{4} P_{2n−2}

C_{4} 2n C_{2n+3}

? ? ? ? cy-

cle with a P_{k}chord P_{k} k = 3 P_{k} k ≥ 4 ? ? ? ? C_{6} C_{6} 3 ≤ p ≤ n− r C_{n}(p, r) n v_{1}, v_{2}, …, v_{n} r v_{1}v_{p}, v_{1}v_{p+1}, …, v_{1}v_{p+r−1}

? ? r = 2, 3 n− 3 n ? ? C_{n}(p, n−

p) p ≡ 0, 3

? C_{n}(3, n−

3)

k_{i} v_{1} v_{i} i = 2, …, n

? ? n v_{1}, v_{2}, …, v_{n} v_{1}v_{k}

v_{1}v_{k+2}(2 ≤

k ≤ n− 3)

? C(n, k)

C_{n} k n k C(n, k)

shelltype C(n, n− 3)

shell C(n, n−

3) F_{n−1} multiple

shell ? H(n, t)

C_{n} t u v u k ≥ 1, n ≥ 4 1 ≤ t ≤ n− 3 G(n, t, k)

k H(n, k) uv G(n, t, k)

t = n− 3 i = 1, 2, …, n v_{i,1}, v_{i,2}, …, v_{i,2m} n C_{2m} ? C_{2m,n} v_{i,m} v_{i+1,m}

i = 1, 2, …, n− 1 C_{6,2k}

C_{8,n}

k n ń ? dragon

? tadpoles

ń C_{n} n ≡ ? ? ń ? n nC_{5} n n

C_{n}^{(t)} t n ? ?

C_{3}^{(t)}

t t ≡ ?

C_{3}^{(t)}

t ≢ 2 ?

C_{n}^{(t)} nt ≡ 0 3 ? t = 2 n ? m ≡ 0 C_{m} α ?? m ≡ 2 C_{m} α ? ?

C_{4}^{(t)}

t ? C_{n} n ≡ 1 ? G G ? tri- an- gu-

lar cactus triangu-

lar snake v_{1}, v_{2}, …, v_{n} v_{i} v_{i+1} w_{i} i = 1, 2, …, n− 1 ? t ≡ t ≡ ? ? n ≡ 2 ? n n ? n ≡ 0 (n ≥

8) n ≡ 2 ? ? K_{4} ? ? ? ? ? ? ? ? ? ? ? ? n ≥ 3 t 1 ≤ t ≤ n C_{n}^{+t}

t n ? n t C_{n}^{+t}

? n t C_{n}^{+t}

? ? ń C_{n} n ≡ 1 ? P_{m}× P_{n} ? ? P_{m}× P_{2} ? P_{m}× P_{2}, m > 2 ? P_{m}× P_{n}

(m, n) ≠ (2, 2) ? ? ? ? Mongolian tent P_{m}× P_{n} n P_{m}× P_{n} Mongolian village Young

tableau P_{m}× P_{n} P_{m}× P_{n} ? ? Prisms C_{m}× P_{n} ? C_{m}× P_{2} m ≡ 0 (mod 4).

? m ? m ? ? C_{m}× P_{n} m n m ≡ 0 C_{4n+2}×

P_{4m+3}

? C_{n}× P_{2} ? C_{6}×

P_{m}(m ≥ 2)

? ? C_{3}× P_{n} n ? C_{m}× P_{n} n C_{4}× P_{2} ? C_{m}× P_{2} m ≠ 4 ? C_{4}× P_{n} n ≥ 3 ? C_{m}× P_{n} n m n 3 ≤ n ≤ 12 m ? C_{2m+1}×

P_{5}

C_{m}× C_{n}

(m > 2, n > 2)

C_{m}× C_{n} m n ? C_{m}× C_{n} m ≡ 1, 2, 3 n ? C_{m}× C_{n} m ≡ 0 n ? C_{m}× P_{2} Möbius lad- der M_{n} P_{n}× P_{2} P_{n} ? ö M_{3} ? C_{m}× P_{2} C_{m}× P_{2}

book B_{m} S_{m}× P_{2} S_{m} m+ 1 ? B_{2m} B_{4m+1}

B_{4m+3}

? ? L_{n}× P_{2} B_{2m}× P_{2} ? ? B_{2m} B_{4m+3}

? ? B_{4m+1}

? B_{4m+1}

?? ? ? stacked book S_{m}× P_{n} S_{2m}× P_{n} S_{2m+1}×

P_{n} n- cube K_{2}× K_{2}× ⋯× K_{2} n ? ? ? K_{4}× P_{n} ? K_{m}× P_{n}

(m, n) = (4, 2), (4, 3), (4, 4), (4, 5) (3, 3) (6, 2).

K_{5}× P_{2} W_{n} n = 3 n G_{1}[G_{2}]

V (G_{1})× V (G_{2}) {(x_{1}, y_{1}), (x_{2}, y_{2})|x_{1}x_{2} ∈

E(G_{1})x_{1} = x_{2}y_{1}y_{2} ∈

E(G_{2})}

G_{1}⊕ G_{2} G_{1} G_{2}

V (G_{1})× V (G_{2}) {(x_{1}, y_{1}), (x_{2}, y_{2})|x_{1}x_{2} ∈ E(G_{1})y_{1}y_{2} ∈ E(G_{2})}

(? P_{n}⊕) / (\overline{K_{2}} n > 1) P_{n}[P_{2}] n C_{m}⊕ C_{n}

C_{m}[C_{n}] m n K_{n} n ≤ 4 ? ? ? ? ? ? K_{m,n}

? m n = 1 ? K_{1,m,n}

? K_{1,1,m,n}

K_{2,m,n}

? m, n p

K_{m,n,p}

? K_{n}− e K_{n} n ≤ 5 K_{n}−

2e(K_{n}

n ≤ 6 K_{n}− 3e n ≤ 6 K_{n}− G G K_{1,a}

a ≤ n− 2 G M_{a} 2a ≤ n K_{1,m,n}, K_{2,m,n}, K_{1,1,m,n}

K_{m,n} windmill K_{n}^{(m)}(n >

3) m K_{n}

K_{n}^{(m)} n ≤ 5 ? ?

K_{4}^{(m)}

m ≥ 4 m ≤ 22 ? ?

K_{n}^{(m)} n = 4 m = 2 m = 2 n = 5 ?

K_{4}^{(m)} m ?

K_{n}^{(2)} n = 4 n n = 6 ?

K_{n}^{(2)} n = 2^{a}p_{1}^{a_{1}} ⋯ p_{s}^{a_{s}} a, a_{1}, …, a_{s} p_{1}, …, p_{s}

j p_{j} ≡

3 a_{j}

K_{n}^{(3)} n ≡ 0 3n = 4^{e}(8k+

7) n ≡ 5 ? ?

K_{m,n}^{(t)},

t K_{m,n}

? K_{2,m_{i}}

i = 1, 2, …, n m_{i} m_{i} ? B(n, r, m) m K_{n} K_{r} n ≥ r ? B(n, r, m) n, r m ? m, n r B(n, r, m) r = 1

K_{n}^{(m)} r > 1, B(n, r, m) n =