Weight Choosability of theta Graphs

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Ting-Feng Jian

Advisor: Gerard Jennhwa Chang

Department of Mathematics, National Taiwan University

August 2, 2014

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Weight Choosability of theta Graphs

Outline

1. Introduction 2. The Paths 3. The Cycles 4. The θ-graphs

5. The Generalized θ-graphs 6. Further Problems

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Weight Choosability of theta Graphs Introduction

Definitions

1. G = (V , E ) be a connected graph but not K2.

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Definitions

2. L(e) ⊆ R, a list of weights of e.

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Definitions

3. L-edge weighting: f such that f (e) ∈ L(e).

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Definitions

4. induced weight: g (v ) =P

uv∈Ef(uv ).

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Definitions

5. proper weighting: g (v ) 6= g (v^′).

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Definitions

1. L(e) = {1, 2, ..., k}, f proper weighting

⇒ G is k-edge weight colorable.

2. L(e) ⊆ R, f proper weighting

⇒ G is k-edge weight choosable.

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Problems

Conjecture (M.Karonski, T.Luczak, and A.Thomason) ([1]) Every connected graph G 6= K2 is 3-edge weight colorable.

Conjecture (T.Bartnicki, J.Grytczuk, and S.Niwczykl) ([2]) Every connected graph G 6= K2 is 3-edge weight choosable.

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Recent Result

Theorem (M.Kalkowski, M.Karonski, and F.Pfender, 2010) ([8]) Every connected graph G 6= K₂ is 5-edge weight colorable.

Theorem (T.Bartnicki, J.Grytczuk, and S.Niwczyk1)

([2]) A clique, complete bipartite graph, or a tree, not K2, is 3-edge weight choosable.

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Polynomials

1. Edge Set: E = {e1, e2, ..., em}.

2. Variables xe = f (e) ∈ L(e).

3. Associated polynomial of G of orientation D:

P_G(x1, x2, ..., xm) = Y

vv^′∈E (D)

X

e=uv ∈E

x_e− X

e^′=u^′v^′∈E

x_e′

! 6= 0

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Example

Then the associated polynomial: PG(x1, x2, x3, x4, x5)

= (x4−x2−x5)(x1+x5−x3)(x2−x4−x5)(x3+x5−x1)(x1+x2−x3−x4).

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Combinatorial Nullstellensatz

Theorem (N. Alon, 1999)

([3]) Let F be an arbitrary field, and let P(x1, x2, ..., xm) be a polynomial in F[x1, x2, ..., xm]. Suppose that the coefficient of x₁^k¹x₂^k²...x_m^k^m in P is non-zero and deg(P) =Pm

i=1k_i. Then for any subsets A1, A2, ..., Am of F satisfying |Ai| ≥ ki + 1 for all i = 1, 2, ..., m, there exists

(a1, a2, ..., am) ∈ A1 × A2× ... × An so that P(a1, a2, ..., am) 6= 0.

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Monomial Index

Define the monomial index by mind(P) = min

M h(M) = min

M max

1≤i ≤mk_i. Coeffiecient of x₁x₂...xm is non-zero

⇒ 2-egde Weight Choosable.

Coeffiecient of x₁^k¹x₂^k²...x_m^k² is non-zero for ki ≤ 2

⇒ 3-egde Weight Choosable.

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Example

Then the associated polynomial: PG(x1, x2, x3, x4, x5)

= (x4−x2−x5)(x1+x5−x3)(x2−x4−x5)(x3+x5−x1)(x1+x2−x3−x4).

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Permanent

Let m × m matrix A = [aij].

1. Permanent:

perA = X

σ∈Sm

m

Y

i=1

a_{i σ(i )}

! .

2. Let K = (k1, k2, ..., km), ki ≥ 0 and Pm

i=1k_i = m.

Repeating the i -th columns ki times, denoted A(K ).

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Permanent Index

permanent index:

The minimum of k so that there is

K = (k1, k2, ..., km), ki ≤ k for all i and perA(K ) 6= 0.

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Orientation

Fixed orientation D of a graph G , define the associated matrix A_G = [aij] by

a_ij =







1, if ej is incident to the head of ei;

−1, if ej is incident to the tail of ei; 0, if ej and ei are not incident.

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The Relation

P_G(x1, x2, x3, x4, x5) A_G

= (x4− x2− x5) 0 −1 0 1 −1 (x₁+ x₅− x₃) 1 0 −1 0 1 (x2− x4− x5) 0 1 0 −1 −1 (x3+ x5− x1) −1 0 1 0 1 (x₁+ x₂− x₃− x₄) 1 1 −1 −1 0

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The Relation

Coefficient of x1x₂x₃x₄x₅: perAG.

Coefficient of x₁²x₂x₃x₄: perAG(2, 1, 1, 1, 0)/2!.

Coefficient of x₁²x₂²x₃: perAG(2, 2, 1, 0, 0)/2!2!.

Coefficient of x₁³x₂x₃: perAG(3, 1, 1, 0, 0)/3!.

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The Lemma

Lemma

([2]) Let A = [aij] be a m × m matrix with finite permanent index. Let the polynomial

P(x1, x2, ..., xm) =

m

Y

i=1 m

X

j=1

a_ijx_j

! ,

then mind(P) = pind(A).

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Useful Result

Theorem

([2]) Let AG be an associated matrix of G . If pind(AG) ≤ k, then G is (k + 1)-edge weight choosable.

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Weight Choosability of theta Graphs The Paths

Paths

Let path Pm : v1v₂...vm+1 with m edges.

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Associated Matrices

A_P_m =







0 −1 0 0 0 . ..

1 0 −1 0 . .. 0

0 1 0 . .. 0 0

0 0 . .. 0 −1 0

0 . .. 0 1 0 −1

. .. 0 0 0 1 0





 .

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2-Choosability

Theorem

Let Pm be a path. Let APm be the associated matrix of Pm, m ≥ 2. Then

perAPm = (−1)^m², if m is even 0, otherwise.

A_P₂ =0 −1 1 0

A_P₃ =





0 −1 0

1 0 −1

0 1 0





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3-Choosability

Lemma

Let APm be the associated matrix of path Pm with m ≥ 4 edges. Let K = (k1, k2, ..., km) where k1 = km = 0,

k₂ = k₃ = 2, and other ki = 1. Then perAPm(K ) = 4 K = (0, 2, 2, 1, ..., 1, 0)

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3-Choosability

A_P_m(K ) =







−1 −1 0 0 0 0 · · · 0 0

0 0 −1 −1 0 0 · · · 0 0

1 1 0 0 −1 0 · · · 0 0

0 0 1 1 0 −1 · · · 0 0

0 0 0 0 1 0 · · · 0 0

... ... ... ... ... ... ... ... ...

0 0 0 0 0 0 · · · 0 −1

0 0 0 0 0 0 · · · 1 0

0 0 0 0 0 0 · · · 0 1





 ,

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Non-2-Choosability

P₃ is 2-choosable. Take x₁, x₃ so that x₁ 6= x₃ and x₂ 6= 0.

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Non-2-Choosability

P_m is not 2-choosable for odd m ≥ 5.

(1) xi 6= xi+2 and x2 6= 0, xm−16= 0.

(2) Assign L(e_2j) = {j − 1, j} for j = 1, 2, ...,^m−3₂

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Weight Choosability of theta Graphs The Cycles

Cycles

Let E = {e₁, e₂, ..., en} be the edge set of Cn. Give the orientation as ei+1 follows ei for i = 1, 2, ..., n − 1 and e1

follows en.

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Associated Matrices

A_C_n =







0 −1 0 0 · · · 0 0 1

1 0 −1 0 · · · 0 0 0

0 1 0 −1 · · · 0 0 0

0 0 1 0 · · · 0 0 0

... ... ... ... . .. ... ... ...

0 0 0 0 · · · 0 −1 0

0 0 0 0 · · · 1 0 −1

−1 0 0 0 · · · 0 1 0





 .

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2-Choosability

n is odd: an= 1ⁿ+ (−1)ⁿ = 0.

n is even:

b_ij =







1, if i − j = 0;

−1, if i − j = −1(mod n);

0, otherwise.

a_n= (1ⁿ² + (−1)ⁿ²)bⁿ₂ = 4, if 4 divides n 0, otherwise.

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2-Choosability

Theorem

Let Cn be a cycle. Let ACn be the associated matrix of Cn, n ≥ 3. Then

perACn = 4, if 4 divides n;

0, otherwise.

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3-Choosability

Theorem

Let ACn be the associated matrix of Cn, n ≥ 4.

Let K = (k1, k2, ..., kn) where k1 = k2 = 2, k3 = k4 = 0 and other ki = 1.

Then

perACn(K ) = (−1)ⁿ× 4.

In particular, perACn(K ) 6= 0.

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Associated Matrices

A_C_n =







0 −1 0 0 · · · 0 0 1

1 0 −1 0 · · · 0 0 0

0 1 0 −1 · · · 0 0 0

0 0 1 0 · · · 0 0 0

... ... ... ... . .. ... ... ...

0 0 0 0 · · · 0 −1 0

0 0 0 0 · · · 1 0 −1

−1 0 0 0 · · · 0 1 0





 .

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Finding A(K )

A_C₄(K ) =







0 0 −1 −1

1 1 0 0

0 0 1 1

−1 −1 0 0







A_C₅(K ) =







0 0 −1 −1 1

1 1 0 0 0

0 0 1 1 0

0 0 0 0 −1

−1 −1 0 0 0





 a₄ = 4, a₅ = −4.

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Finding A(K )

A_C_n(K ) =







0 0 −1 −1 0 · · · 0 1

1 1 0 0 0 · · · 0 0

0 0 1 1 0 · · · 0 0

0 0 0 0 −1 · · · 0 0

0 0 0 0 0 · · · 0 0

... ... ... ... ... . .. ... ...

0 0 0 0 0 · · · 0 −1

−1 −1 0 0 0 · · · 1 0





 .

a_n = a_n−2 = (−1)ⁿ× 4.

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Non-2-Choosability

Theorem

If 4 does not divide n, then Cn is not 2-edge weight colorable.

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Consequences

2-edge weight choosable:

P₃, Pm for even m and Cn for 4 divides n.

3-edge weight choosable:

P_m for odd m 6= 3 and Cn for 4 does not divide n.

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Weight Choosability of theta Graphs The θ-graphs

What Is a θ-graph?

θ(m₁, m₂, m₃) for the θ-graph if the lengths of the upper, middle, and lower paths are m1, m2, m3, respectively.

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Associated Matrices

A_θ(3,2,3) =







0 −1 0 1 0 1 0 0

1 0 −1 0 0 0 0 0

0 1 0 0 −1 0 0 −1

1 0 0 0 −1 1 0 0

0 0 −1 1 0 0 0 −1

1 0 0 1 0 0 −1 0

0 0 0 0 0 1 0 −1

0 0 −1 0 −1 0 1 0





 .

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Associated Matrices





A_X A_XY A_XZ A_YX A_Y A_YZ A_ZX A_ZY A_Z



,

A_X, AY, and AZ: associated matrix of paths of lengths m₁, m2, m3, respectively.

Other submatrices have only two numbers: 1 on the upper left and −1 on the lower right.

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Notations

1. S = (R, C ) where |R| = |C | and

R ⊆ {1, 2, ..., m}, C ⊆ {1, 2, ..., m}.

2. AS: submatrix of A formed by the R rows and C -th columns.

3. A^(S): submatrix of A obtained by deleting the R rows and C columns.

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The Main Proposition

Proposition

Let A_θ(m₁,m₂,m₃) be the associated matrix of θ(m1, m2, m3).

Let S3 demote the permutation group of rank 3. Then 1. perAθ(m1,m₂,m₃)= perAθ(mσ(1),mσ(2),mσ(3)) for all σ ∈ S3. 2. perA_θ(m₁_+4,m₂,m₃)= perA_θ(m₁,m₂,m₃) for m₁ ≥ 3.

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The Proof

Let A = A_θ(m₁_+4,m₂,m₃) and B = A_θ(m₁,m₂,m₃). Such A has the following form:

A=







0 −1 0 0 0 0 0 · · ·

1 0 −1 0 0 0 0 · · ·

0 1 0 −1 0 0 0 · · ·

0 0 1 0 −1 0 0 · · · AXY A_XZ

0 0 0 1 0 −1 0 · · ·

0 0 0 0 1 0 −1 · · ·

0 0 0 0 0 1 0 · · ·

... ... ... ... ... ... ... . .. ... ...

A_YX A_Y A_YZ

A_ZX A_ZY A_Z





 .

Choose R = {2, 3, 4, 5, 6} with |R| = 5.

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The Proof

perAS1 = perA(R,{1,2,3,4,5}) = per







1 0 −1 0 0

0 1 0 −1 0

0 0 1 0 −1

0 0 0 1 0

0 0 0 0 1







= 1.

perAS₂ = perA(R,{1,3,4,5,6}) = per







1 −1 0 0 0

0 0 −1 0 0

0 1 0 −1 0

0 0 1 0 −1

0 0 0 1 0







= 1.

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The Proof

perAS3 = perA(R,{1,2,3,4,7}) = per







1 0 −1 0 0

0 1 0 −1 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 −1







= 0.

perAS₄ = perA(R,{2,3,4,5,6}) = per







0 −1 0 0 0

1 0 −1 0 0

0 1 0 −1 0

0 0 1 0 −1

0 0 0 1 0







= 0.

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The Proof

perAS5 = perA(R,{2,3,4,5,7}) = per







0 −1 0 0 0

1 0 −1 0 0

0 1 0 −1 0

0 0 1 0 0

0 0 0 1 −1







= −1.

perAS₆ = perA(R,{3,4,5,6,7}) = per







−1 0 0 0 0

0 −1 0 0 0

1 0 −1 0 0

0 1 0 −1 0

0 0 1 0 −1







= −1.

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The Proof

perA =

6

X

k=1

perASkperA^(S^k⁾

= perA^(S¹⁾+ perA^(S²⁾− perA^(S⁵⁾− perA^(S⁶⁾.

perB =

m1+m2+m3

X

k=1

perB_({2},{k})perB^({2},{k})

= perB^({2},{1})− perB^({2},{3}).

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The Proof

perA^(S¹⁾+ perA^(S²⁾= perB^({2},{1}), perA^(S⁵⁾+ perA^(S⁶⁾ = perB^({2},{3})

perA = perB.

perAθ(m1+4,m2,m₃) = perAθ(m1,m₂,m₃).

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The Table of perA

θ(m1,m2,m3)

m₁ = 1 m3 = 2 m3 = 3 m3 = 4 m3 = 5 m3 = 6

m₂ = 2 0 4 0 4 0

m₂ = 3 0 −4 0 4

m₂ = 4 0 −4 0

m₂ = 5 0 4

m₂ = 6 0

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The Table of perA

θ(m1,m2,m3)

m₁ = 2 m3 = 2 m3 = 3 m3 = 4 m3 = 5 m3 = 6

m₂ = 2 −20 0 4 0 −20

m₂ = 3 4 0 4 0

m₂ = 4 −4 0 4

m₂ = 5 4 0

m₂ = 6 −20

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The Table of perA

θ(m1,m2,m3)

m₁ = 3 m3 = 3 m3 = 4 m3 = 5 m3 = 6

m₂ = 3 0 −4 0 4

m₂ = 4 0 −4 0

m₂ = 5 0 4

m₂ = 6 0

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The Table of perA

θ(m1,m2,m3)

m₁ = 4 m₃ = 4 m₃ = 5 m₃ = 6

m₂ = 4 20 0 −4

m₂ = 5 −4 0

m₂ = 6 4

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The Table of perA

θ(m1,m2,m3)

m₁ = 5 m3 = 5 m3 = 6

m₂ = 5 0 4

m₂ = 6 0

m₁ = 6 m3 = 6 m₂ = 6 −20

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2-Choosability

Theorem

Let Aθ(m1,m₂,m₃) be the associated matrix of θ(m1, m2, m3).

Then perA_θ(m₁,m₂,m₃) 6= 0 if and only if m = m₁ + m₂+ m₃ is even.

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Useful Proposition

Proposition

([2]) Let G be a graph whose edge set can be partitioned into two subgraph P, Q, in which P = {e1, e2, ..., em}.

Assume that the associated matrices AP, AQ have permanent indexes at most 2. Let perAP(K ) 6= 0 where K = (k₁, k₂, ..., km) with ki = 0 for any correspondent edge e_i incident to Q. Then pind(AG) ≤ 2.

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The Proof

We can separate P into two parts:

P₁ = {ei ∈ P : ei does not link to Q}, P₂ = {ei ∈ P : ei link to Q}.

A_G =





A_P₁ ... 0 ... A_P₂ ...

0 ... A_Q



.

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The Proof

By assumption, all the edges ei in P2 gives ki = 0.

A_G(K^′) =AP(K ) ...

0 A_Q(K^(Q))

with permanent

perAG(K^′) = perAP(K ) × perAQ(K^(Q)) 6= 0.

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Consequences

m₁ ≤ m2 ≤ m3.

If m3 ≥ 4, then P = Pm₃ and Q = Cm₁+m2. Check the case m3 ≤ 3.

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3-Choosability

m₁ m₂ m₃ K perA_θ(m₁,m₂,m₃)(K )

1 2 2 (0, 2, 1, 1, 1) 12

1 2 3 (1, 1, 1, 1, 1, 1) 4

1 3 3 (2, 0, 1, 1, 1, 1, 1) 16 2 2 2 (1, 1, 1, 1, 1, 1) −20 2 2 3 (2, 1, 0, 1, 1, 1, 1) −4 2 3 3 (1, 1, 1, 1, 1, 1, 1, 1) 4 3 3 3 (2, 2, 1, 0, 0, 1, 1, 1, 1) 4

.

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Weight Choosability of theta Graphs The Generalized θ-graphs

What Is a Generalized θ-graph?

Generalized θ-graph θ(m1, m2, ..., mp):

p paths which have the two common endpoints.

In particular, θ(m) = Pm and θ(m₁, m₂) = Cm₁+m2.

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A Useful Theorem

Theorem

([2]) If G 6= K₂ is a clique, complete bipartite graph, or a tree, then mind(G ) ≤ 2.

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Step 1

Step 1. θ(2, 2, ..., 2) = K2,p is 3-edge weight choosable.

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Step 2

Step 2. mi ≥ 2 for all i = 1, 2, ...p

θ-graph θ(m1, m2, ..., mp) with is 3-edge weight choosable.

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Step 3

Step 3. θ(m1, m2, ..., mp, 1) is 3-edge weight choosable.

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Step 3

Let L ⊆ R and c ∈ R.

L+ c = {l + c : l ∈ L}.

Arbitrary choose x ∈ L(u0u₁) and fix this x. p ≥ 3, define a lists L^′(e) on θ(m1, m2, ..., mp) by

L^′(e) = L(e) + x p− 2.

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Odd Cycle Absorbs P

3

Theorem

Assume that k ≥ 3. Let G = (V , E ) be a graph. Suppose there are path P3 = v0v₁v₂v₃ and odd cycle Ct in G such that P3∩ Ct = {v0, v3} ⊂ V . If G − P3 is k-edge weight choosable, then so is G .

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The Proof

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The Proof

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The Proof

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The Proof

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The Proof

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The Proof

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The Proof

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The Proof

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The Proof

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The Proof

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The Proof

L^′(e) =











L(e) + ^x^v0v1₂ +^x^v2v3₂ , if e = ui−1u_i for odd i ≤ s;

L(e) −^x^v0v1₂ − ^x^v2v3₂ , if e = ui−1ui for even i < s;

L(e) + ^x^v0v1₂ − ^x^v2v3₂ , if e = ui−1u_i for odd i > s;

L(e) −^x^v0v1₂ + ^x^v2v3₂ , if e = ui−1u_i for even i > s;

L(e), otherwise.

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The Proof

x_e =











x_e^′ − ^x^v0v1₂ − ^x^v2v3₂ , if e = ui−1u_i for odd i ≤ s;

x_e^′ + ^x^v0v1₂ +^x^v2v3₂ , if e = ui−1ui for even i < s;

x_e^′ − ^x^v0v1₂ +^x^v2v3₂ , if e = ui−1u_i for odd i > s;

x_e^′ + ^x^v0v1₂ −^x^v2v3₂ , if e = ui−1u_i for even i > s;

x_e^′, otherwise

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Weight Choosability of theta Graphs Further Problems

Problems

1. Total Weight Choosability, by T. Wong and X. Zhu [9].

2. Weight Choosability of Hypergraphs,

by M. Kalkowski, M. Karonski, and F. Pfender [10].

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Weight Choosability of theta Graphs References

References

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References

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References

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References

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