Long properly colored cycles in edge-colored complete graphscaagt.ugent.be/LPLC/Li.pdf · Long...

Long properly colored cycles in edge-colored completegraphs

Ruonan Li a,b

Joint work with

Hajo Broersma b, Chuandong Xu a and Shenggui Zhang a

a Northwestern Polytechnical University, Xi’an, Chinab University of Twente, Enschede, Netherlands

Email: [email protected]; [email protected]

Ruonan Li (NPU&UT) Long properly colored cycles in edge-colored complete graphs 1 / 19

Backgrounds Terminology and Notation

G = (V,E): a finite, simple and undirected graph

edge-coloring: a mapping col : E→ N

edge-colored graph: a graph assigned an edge-coloring

col(e),col(H),col(F,H),col(v,H).

properly colored (PC) graph: a colored graph in which each pairof adjacent edges has distinct colors

dc(v): the number of colors of the edges incident to v, called thecolor degree of v in G.

δ c(G) = min{dc(v) : v ∈V (G)}: the minimum color degree of G.

Gi: the subgraph of G induced by the edges of color i.∆mon(G): the maximum monochromatic degree of G, i.e,

∆mon(G) = max{∆(Gi) : i ∈ col(G)}

Gi: the subgraph of G induced by the edges of color i.

∆mon(G): the maximum monochromatic degree of G, i.e,

Directed cycles and PC cycles

Figure: Transition.

Backgrounds Conjecture

Conjecture 1 (Bollobas and Erdos, IJM, 1976)

Every edge-colored Kn with ∆mon(Kn)< bn2c contains a properly colored

Hamiltonian cycle.

Find the maximum p such that every Kcn with ∆mon(Kc

n)< pncontains a properly colored Hamilton cycle.

p≥ 1/64 (Bollobas and Erdos, 1976)p≥ 1/17 (Chen,1976)p≥ 1/7 (Shearer, 1979)

When n is sufficently largep≥ 1−1/

√2− ε (Alon and Gutin, 1997)

p≥ 1/2− ε (Lo, 2016+)

Hamiltonian cycle.

p≥ 1/2− ε (Lo, 2016+)

Hamiltonian cycle.

p≥ 1/2− ε (Lo, 2016+)

Hamiltonian cycle.

p≥ 1/64 (Bollobas and Erdos, 1976)

p≥ 1/17 (Chen,1976)p≥ 1/7 (Shearer, 1979)

p≥ 1/2− ε (Lo, 2016+)

Hamiltonian cycle.

p≥ 1/64 (Bollobas and Erdos, 1976)p≥ 1/17 (Chen,1976)

p≥ 1/7 (Shearer, 1979)

p≥ 1/2− ε (Lo, 2016+)

Hamiltonian cycle.

p≥ 1/2− ε (Lo, 2016+)

Hamiltonian cycle.

When n is sufficently large

p≥ 1−1/√

2− ε (Alon and Gutin, 1997)p≥ 1/2− ε (Lo, 2016+)

Hamiltonian cycle.

p≥ 1/2− ε (Lo, 2016+)

Hamiltonian cycle.

p≥ 1/2− ε (Lo, 2016+)

In 2011, Fujita and Magnant 1 conjectured the following.

Conjecture 2 (Fujita and Magnant, DAM, 2011)

If δ c(Kn)≥ n+12 , then each vertex of Kn is contained in a PC cycle of

length k for all 3≤ k ≤ n.

A colored K2m is an extremal graph.

Figure: An example when δ c(K6) = 3

1S. Fujita and C. Magnant, Properly colored paths and cycles, DiscreteAppl. Math., 159 (2011) 1391–1397.

Theorem 1 (Fujita and Magnant, DAM, 2011)

If δ c(Kn)≥ n+12 . Then each vertex of Kn is contained in a PC C3 and a

PC C4. If n≥ 13, then each vertex of Kn is contained in a PC cycle oflength at least 5.

Problem 1

Determine the largest value of f (n) such that each vertex of Kn iscontained in a PC cycle of length at least f (n) when δ c(Kn)≥ n+1

A PC Hamiltonian cycle ⇒ f (n) = n.

By Theorem 1, f (n)≥ 4, and f (n)≥ 5 when n≥ 13.

Problem 1

Our results Main theorem

Theorem 2 (Li, Broersma, Xu and Zhang, 2016+)

length at least δ c(Kn), i.e., f (n)≥ δ c(Kn).

PC cycle extension theorem

Let G be a colored Kn, and let C be a PC cycle of length k in G. Ifδ c(G)≥max{n−k

2 ,k}+1, then G contains a PC cycle C∗ such thatV (C)⊂V (C∗) and |C∗|> k.

PC path lemma

Lemma 1 (Li, Broersma, Xu and Zhang, 2016+)

Let G be a colored Kn, let C be a PC cycle of length k in G, and letx,y be two distinct vertices in V (G)\V (C). If x→C, then there is aPC path of length k+1 starting at x, ending at y and passing throughall vertices of V (C).

x→C: ∃u ∈V (C) s.t. col(xu) 6∈ {col(uu+),col(uu−)}.

PC path lemma

Our results Proof techniques

Proof: By contradiction. Assume that G is a colored Kn withδ c(G)≥max{n−k

2 ,k}+1 and C is a PC cycle of length k in G.Suppose that there are no PC cycles longer than and containing C.Let S =V (G)\V (C).

Claim 1

There exist vertices u∗ ∈V (C) and v∗ ∈ S such that col(u∗v∗) 6∈ col(C).

Proof: By contradiction. Assume that G is a colored Kn withδ c(G)≥max{n−k

2 ,k}+1 and C is a PC cycle of length k in G.Suppose that there are no PC cycles longer than and containing C.Let S =V (G)\V (C).

Claim 1

There exist vertices u∗ ∈V (C) and v∗ ∈ S such that col(u∗v∗) 6∈ col(C).

Claim 2

For each vertex v ∈ S, there exists a vertex v′ ∈ S such thatcol(vv′) 6∈ col(v,C).

v1(=v∗)v2 v3

Lemma 1 forces that col(v1v2) ∈ col(v2,C)

Lemma 1 and Claim 2 force that v1v2v3 is a PC path

Repeat using Claim 2 and Lemma 1

Claim 2

v1(=v∗)v2 v3

Claim 2

v1(=v∗)v2 v3

Claim 2

v1(=v∗)v2 v3

. . .vl

v1(=v∗)v2 v3

. . .vj

vj+1. . .vl−1

Figure: Two situations when adding a new vertex v′

If case (a) happens, then continue;Otherwise, set s = l; output: u∗,v1,v2, . . . ,vs.

v1(=v∗)v2 v3

. . .vl

v1(=v∗)v2 v3

. . .vj

vj+1. . .vl−1

If case (a) happens, then continue;

Otherwise, set s = l; output: u∗,v1,v2, . . . ,vs.

v1(=v∗)v2 v3

. . .vl

v1(=v∗)v2 v3

. . .vj

vj+1. . .vl−1

If case (a) happens, then continue;Otherwise, set s = l; output: u∗,v1,v2, . . . ,vs.

Let u∗,v1,v2, . . . ,vs be an output with least number of vertices.

Claim 3

(a) vs = v1.(b) col(vi,C) = {col(vi−1vi)} for all i with 1≤ i≤ s−1.

CC ′

vi−1

v1(=vs)

v∗(=v1) satisfies that |col(v∗,C)|= 1 and col(v∗,C)∩ col(C) 6= /0.

Claim 3

CC ′

vi−1

v1(=vs)

Claim 3

CC ′

vi−1

v1(=vs)

Claim 3

CC ′

vi−1

v1(=vs)

X = {u ∈ S : |col(u,C)|= 1 and col(u,C)∩ col(C) = /0},

Y = {u ∈ S : |col(u,C)|= 1, and col(u,C)∩ col(C) 6= /0},Z = {u ∈ S : |col(u,C)| ≥ 2}.Define col(u) = col(u,C) for each vertex u ∈ X ∪Y .

Yα = {y ∈ Y : ∃z ∈ Z with col(zy) 6= col(y) or∃y′ ∈ Y with col(yy′) 6∈{col(y),col(y′)}},Yβ = Y \Yα .

X Yα Yβ Z

X = {u ∈ S : |col(u,C)|= 1 and col(u,C)∩ col(C) = /0},Y = {u ∈ S : |col(u,C)|= 1, and col(u,C)∩ col(C) 6= /0},

Z = {u ∈ S : |col(u,C)| ≥ 2}.Define col(u) = col(u,C) for each vertex u ∈ X ∪Y .

X Yα Yβ Z

X = {u ∈ S : |col(u,C)|= 1 and col(u,C)∩ col(C) = /0},Y = {u ∈ S : |col(u,C)|= 1, and col(u,C)∩ col(C) 6= /0},Z = {u ∈ S : |col(u,C)| ≥ 2}.

Define col(u) = col(u,C) for each vertex u ∈ X ∪Y .

X Yα Yβ Z

X = {u ∈ S : |col(u,C)|= 1 and col(u,C)∩ col(C) = /0},Y = {u ∈ S : |col(u,C)|= 1, and col(u,C)∩ col(C) 6= /0},Z = {u ∈ S : |col(u,C)| ≥ 2}.Define col(u) = col(u,C) for each vertex u ∈ X ∪Y .

X Yα Yβ Z

Yα = {y ∈ Y : ∃z ∈ Z with col(zy) 6= col(y) or∃y′ ∈ Y with col(yy′) 6∈{col(y),col(y′)}},

Yβ = Y \Yα .

X Yα Yβ Z

Claim 4

(a) col(uv) ∈ {col(u),col(v)} for all distinct vertices u,v ∈ X ∪Yβ .

(b) col(uv) = col(u) for all vertices u ∈ X ∪Yβ and v ∈V (C)∪Z∪Yα .

tournament

X ∪ Yβw∗

d−D(w∗) ≥ |X|+|Yβ |−1

C Yα Z

col(w∗) appears at least|X |+|Yβ |−1

2 + |C|+ |Yα |+ |Z| times at w∗.

Claim 4

(a) col(uv) ∈ {col(u),col(v)} for all distinct vertices u,v ∈ X ∪Yβ .

(b) col(uv) = col(u) for all vertices u ∈ X ∪Yβ and v ∈V (C)∪Z∪Yα .

tournament

X ∪ Yβw∗

d−D(w∗) ≥ |X|+|Yβ |−1

C Yα Z

2 + |C|+ |Yα |+ |Z| times at w∗.

Claim 4

(a) col(uv) ∈ {col(u),col(v)} for all distinct vertices u,v ∈ X ∪Yβ .(b) col(uv) = col(u) for all vertices u ∈ X ∪Yβ and v ∈V (C)∪Z∪Yα .

tournament

X ∪ Yβw∗

d−D(w∗) ≥ |X|+|Yβ |−1

C Yα Z

2 + |C|+ |Yα |+ |Z| times at w∗.

Claim 4

(a) col(uv) ∈ {col(u),col(v)} for all distinct vertices u,v ∈ X ∪Yβ .(b) col(uv) = col(u) for all vertices u ∈ X ∪Yβ and v ∈V (C)∪Z∪Yα .

tournament

X ∪ Yβw∗

d−D(w∗) ≥ |X|+|Yβ |−1

C Yα Z

2 + |C|+ |Yα |+ |Z| times at w∗.

Since d(w∗) = n−1, we deduce

n−1≥ dc(w∗)+|X |+ |Yβ |−1

2+ |C|+ |Z|+ |Yα |−1.

Using n = |X |+ |Yα |+ |Yβ |+ |C|+ |Z|, this yields

dc(w∗)≤ |X |+ |Yβ |+12

or, equivalently,

|X |+ |Yβ | ≥ 2dc(w∗)−1≥ 2δc(G)−1≥ 2(

n−|C|2

+1)−1 = n−|C|+1.

This implies that |X |+ |Yβ |+ |C| ≥ n+1, a contradiction.

n−1≥ dc(w∗)+|X |+ |Yβ |−1

2+ |C|+ |Z|+ |Yα |−1.

dc(w∗)≤ |X |+ |Yβ |+12

or, equivalently,

|X |+ |Yβ | ≥ 2dc(w∗)−1≥ 2δc(G)−1≥ 2(

n−|C|2

+1)−1 = n−|C|+1.

n−1≥ dc(w∗)+|X |+ |Yβ |−1

2+ |C|+ |Z|+ |Yα |−1.

dc(w∗)≤ |X |+ |Yβ |+12

or, equivalently,

|X |+ |Yβ | ≥ 2dc(w∗)−1≥ 2δc(G)−1≥ 2(

n−|C|2

+1)−1 = n−|C|+1.

n−1≥ dc(w∗)+|X |+ |Yβ |−1

2+ |C|+ |Z|+ |Yα |−1.

dc(w∗)≤ |X |+ |Yβ |+12

or, equivalently,

|X |+ |Yβ | ≥ 2dc(w∗)−1≥ 2δc(G)−1≥ 2(

n−|C|2

+1)−1 = n−|C|+1.

Remarks and Questions

Remark 1: When k ≥ n/3, then max{n−k2 ,k}= k.

Corollary 1 (Li, Broersma, Xu and Zhang, 2016+)

Let C be a PC cycle of length k in a colored Kn. If k ≥ n3 , then there

exists a PC cycle C∗ of length at least δ c(Kn) such that V (C)⊆V (C∗).

Question 1

Let C be a PC cycle in a colored Kn. Does there exist a PC C′ with|C′| ≥ δ c(Kn) and V (C)⊆V (C′) ? Here, C′ is not necessarily distinctfrom C.

Remark 1: When k ≥ n/3, then max{n−k2 ,k}= k.

Corollary 1 (Li, Broersma, Xu and Zhang, 2016+)

Let C be a PC cycle of length k in a colored Kn. If k ≥ n3 , then there

exists a PC cycle C∗ of length at least δ c(Kn) such that V (C)⊆V (C∗).

Question 1

Let C be a PC cycle in a colored Kn. Does there exist a PC C′ with|C′| ≥ δ c(Kn) and V (C)⊆V (C′) ? Here, C′ is not necessarily distinctfrom C.

Remark 2: Weaken the conclusion (no“each vertex”).

Question 2

Does every colored Kn contains a PC cycle of length at leastδ c(Kn)+1 when δ c(Kn)≥ 2 ?

By a result of Yeo 2, when δ c(Kn) = 2, the answer is Yes.

Theorem 4 (Yeo, JCTB, 1997)

Let G be an edge-colored graph containing no PC cycles. Then thereis a vertex z ∈V (G) such that no component of G− z is joint to z withedges of more than one color.

2A. Yeo, A note on alternating cycles in edge-colored graphs,J. Combin.Theory Ser. B, 69 (1997) 222–225.

Remark 2: Weaken the conclusion (no“each vertex”).

Question 2

Does every colored Kn contains a PC cycle of length at leastδ c(Kn)+1 when δ c(Kn)≥ 2 ?

By a result of Yeo 2, when δ c(Kn) = 2, the answer is Yes.

Theorem 4 (Yeo, JCTB, 1997)

Let G be an edge-colored graph containing no PC cycles. Then thereis a vertex z ∈V (G) such that no component of G− z is joint to z withedges of more than one color.

2A. Yeo, A note on alternating cycles in edge-colored graphs,J. Combin.Theory Ser. B, 69 (1997) 222–225.

By a result of Lo 3, when δ c(Kn)≥ n+12 , the answer is Yes.

Theorem 5 (Lo, JGT, 2014)

Every edge-colored graph G with δ c(G)≥ 2 contains a PC path oflength 2δ c(G) or a PC cycle of length at least δ c(G)+1.

How about the case 3≤ δ c(Kn)≤ n+12 ?

3A. Lo, A Dirac type condition for properly coloured paths and cycles,Journal of Graph Theory 76 (2014) 60–87.

By a result of Lo 3, when δ c(Kn)≥ n+12 , the answer is Yes.

Theorem 5 (Lo, JGT, 2014)

Every edge-colored graph G with δ c(G)≥ 2 contains a PC path oflength 2δ c(G) or a PC cycle of length at least δ c(G)+1.

How about the case 3≤ δ c(Kn)≤ n+12 ?

3A. Lo, A Dirac type condition for properly coloured paths and cycles,Journal of Graph Theory 76 (2014) 60–87.

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