fractional edge coloring

Fractional EdgeColoring

Debbie Berg

Fractional EdgeChromaticNumber

Hypergraphs andCoveringNumbers

Matchings

Edge ChromaticNumber

Fractional EdgeColorings

Theorems

Transversals andMatchings

Bounds on χ′and χ′

Fractional Edge Coloring

Debbie Berg

Matchings

Theorems

Material from

1 Fractional Graph Theory: A Rational Approach to the Theory ofGraphs, by Edward Scheinerman and Daniel Ullman

2 Stephen Hartke

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Matchings

Theorems

Definitions (Review)

� A hypergraph H is a pair (S ,X ) where S is a finite set and Xconsists of subsets of S , called hyperedges.

� Note that a graph is a hypergraph where we restrict the subsetsto size 2.

� A covering of H is a collection of hyperedges such that everyelement of S is contained in some hyperedge.

� The covering number k(H) of a hypergraph H is the number ofhyperedges in the smallest covering of H.

� A t-fold covering of H is a collection of hyperedges such thatevery element of S is contained in at least t hyperedges.

� The t-fold covering number kt(H) is the number of hyperedgesin the smallest t-fold covering of H.

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Matchings

Theorems

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Matchings

Theorems

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Matchings

Theorems

Let H(G ) be the hypergraph whose vertices are the edges of G andwhose hyperedges are the maximal matchings of G .

Example:

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Matchings

Theorems

Example:

Matchings of G

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Matchings

Theorems

Example:

G and H(G )

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Matchings

Theorems

Fractional Covering Number (Review)

� Recall that the (t-fold) covering number is the number ofhyperedges in the smallest (t-fold) covering of H.

� We similarly define the fractional covering number kf (H) aslimt→∞

� Equivalently, kf = inftkt

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Matchings

Theorems

Matchings

� Recall that a matching is a collection of edges such that novertex is incident to more than one edge in the matching. Thematching number, α′(G ), is the greatest number of edges in amatching of G .

� In integer programming terms, we let xe ∈ {0, 1} be theindicator variable that is 1 if e ∈ M and 0 otherwise. Then

α′(G ) = max∑

xe such that ∀v ,∑e3v

xe ≤ 1.

� To find the fractional matching number, α′∗(G ), simply relaxthe requirements to 0 ≤ xe and xe ≤ 1. This gives us the desiredlinear program.

� Note that since the fractional matching number has fewerrequirements and this is a maximum, α′(G ) ≤ α′∗(G ).

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Matchings

Theorems

Matchings

α′(G ) = max∑

xe ≤ 1.

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Matchings

Theorems

Matchings

α′(G ) = max∑

xe ≤ 1.

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Matchings

Theorems

Matchings

α′(G ) = max∑

xe ≤ 1.

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Matchings

Theorems

Vertex Covers

� The fractional vertex cover number, β∗(G ), is simply the dual ofα′∗(G ). In other words, if 0 ≤ xv , xv ≤ 1,

β∗(G ) = min∑

xv such that ∀e,∑v∈e

xv ≥ 1.

� The traditional vertex cover number, β(G ) is the same thing,with the restriction that xv ∈ {0, 1}. Since this is a minimumand β has more restrictions, β∗(G ) ≤ β(G ).

� Note that LP duality gives us that α′∗ = β∗, so

α′ ≤ α′∗ = β∗ ≤ β.

� For bipartite graphs, the Konig-Egervary Theorem states thatα′ = β, so equality holds throughout.

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Matchings

Theorems

Vertex Covers

β∗(G ) = min∑

xv ≥ 1.

α′ ≤ α′∗ = β∗ ≤ β.

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Matchings

Theorems

Vertex Covers

β∗(G ) = min∑

xv ≥ 1.

α′ ≤ α′∗ = β∗ ≤ β.

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Matchings

Theorems

Vertex Covers

β∗(G ) = min∑

xv ≥ 1.

α′ ≤ α′∗ = β∗ ≤ β.

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Matchings

Theorems

Examples

� α′(C5) = 2

� α′∗(C5) = 2.5

� α′(C6) = α′∗(C6) = 3, since C6 is bipartite

� Note that in C5, the 1/2 matching isn’t a convex combination ofinteger matchings, but in C6, it is.

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Matchings

Theorems

Examples

� α′(C5) = 2

� α′∗(C5) = 2.5

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Matchings

Theorems

Examples

� α′(C5) = 2

� α′∗(C5) = 2.5

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Matchings

Theorems

Examples

� α′(C5) = 2

� α′∗(C5) = 2.5

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Matchings

Theorems

Partitions into Matchings

� There are many ways to partition the edge set of a graph G intomatchings.

� Here are three such ways for the graph G from earlier.

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Matchings

Theorems

Partitions into Matchings

� There are many ways to partition the edge set of a graph G intomatchings.

� Here are three such ways for the graph G from earlier.

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Matchings

Theorems

Edge Chromatic Number

� The size of the smallest partition into matchings is called theedge chromatic number, χ′(G ).

� Recall that we usually think of χ′ as being the minimum numberof colors required to properly edge color a graph, but this is thesame thing, since if each part is assigned a color, a propercoloring results.

� Alternatively, the edge chromatic number is the covering numberof H(G ).

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Matchings

Theorems

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Matchings

Theorems

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Matchings

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Here, we need all three hyperedges to contain all vertices of H(G ), soχ′(G ) = 3.

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Matchings

Theorems

Fractional Edge Chromatic Number

� The fractional edge chromatic number, χ′f (G ), is equal to thefractional covering number kf (H(G )).

� Equivalently, χ′f (G ) = χf (L(G )), where χf is the fractionalchromatic number and L(G ) is the line graph of G .

� Additionally,

χ′f (G ) = limt→∞

χ′t(G )

� Finally, and most importantly, χ′f (G ) relates to valid fractionaledge colorings.

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Matchings

Theorems

� Additionally,

χ′t(G )

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Matchings

Theorems

� Additionally,

χ′t(G )

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Matchings

Theorems

� Additionally,

χ′t(G )

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Matchings

Theorems

Fractional Edge Colorings

� A fractional edge coloring assigns non-negative weights wM toeach matching M of G such that for each edge e ∈ E (G ),∑

wM ≥ 1.

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Matchings

Theorems

� A fractional edge coloring assigns non-negative weights wM toeach matching M of G such that for each edge e ∈ E (G ),∑

wM ≥ 1.

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Matchings

Theorems

� This leads to a picture like:

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Matchings

Theorems

� Let w be a fractional edge coloring. Then the fractional edgechromatic number χ′f (G ) is equal to

� By duality,

χ′f (G ) = maxw

where for each matching,∑

we ≤ 1.

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Matchings

Theorems

� Let w be a fractional edge coloring. Then the fractional edgechromatic number χ′f (G ) is equal to

� By duality,

χ′f (G ) = maxw

where for each matching,∑

we ≤ 1.

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Matchings

Theorems

Theorem 2.1.5

For any graph G , 2α′∗ is an integer. Moreover, there exists afractional matching f for which∑

e∈E(G)

f (e) = α′∗(G )

such that f (e) ∈ {0, 1/2, 1} for every edge e.

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Matchings

Theorems

Proof of Theorem 2.1.5

� Of all matchings f such that∑

e f (e) = α′∗, choose one withthe greatest number of edges such that f (e) = 0. Let H be thesubgraph induced on the set of edges with nonzero values.

� We next prove several claims about H.

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Matchings

Theorems

Proof of Theorem 2.1.5

� Of all matchings f such that∑

e f (e) = α′∗, choose one withthe greatest number of edges such that f (e) = 0. Let H be thesubgraph induced on the set of edges with nonzero values.

� We next prove several claims about H.

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Matchings

Theorems

Claim 1

� Claim: H has no even cycles.

� Proof:� Suppose for the sake of contradiction that there exists an even

cycle C . There must be some edge with minimal value; supposethat edge is e0 and f (e0) = m.

� Define g : E(G)→ {−1, 0, 1} such that g(e0) = −1, galternates between −1 and 1 around C , and g is 0 on everyother edge of G .

� Then f + mg is a new fractional matching with the same sumand one more 0 edge, which is a contradiction.

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Matchings

Theorems

Claim 1

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Matchings

Theorems

Claim 1

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Matchings

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Claim 1

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Matchings

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Claim 2

� Claim: H has no pendant edges, unless they’re entirecomponents of H.

� Proof: Suppose e = uv were a pendant edge where d(u) > 1and d(v) = 1.

� If f (e) = 1, then e is an entire component of H, since no edgetouching it can have value other than 0.

� If f (e) < 1, then increasing f (e) to 1 and decreasing the valueof f on all edges incident to u to 0 creates a matching with thesame sum but more 0 edges, a contradiction.

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Matchings

Theorems

Claim 2

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Matchings

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Claim 2

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Matchings

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Claim 2

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Matchings

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Claim 3

� Claim: If H has an odd cycle C , it must be an entirecomponent of H.

� Proof: Suppose C is an odd cycle and suppose v ∈ C is suchthat dH(v) ≥ 3.

� Start at v , choose an edge not in C , and trace a longest path inH. Note that this path can’t return to C , since that wouldcreate an even cycle.

� The path also can’t reach a vertex of degree 1, since there aren’tany pendant edges.

� Therefore, it ends at a vertex such that any other edge revisitsthe path, which would form another cycle.

� Thus, H contains a subset K that consists of two odd cyclesconnected by a path, where the path may have length 0.

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Matchings

Theorems

Claim 3

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Matchings

Theorems

Claim 3

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Matchings

Theorems

Claim 3

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Matchings

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Claim 3

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Claim 3

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Matchings

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Claim 3 (continued)

� H contains a subset K that consists of two odd cycles connectedby a path, where the path may have length 0.

� Let g assign 0 to edges not on K , alternate 1/2 and −1/2around the cycles, and alternate 1 and −1 on the path betweenthem. Note that this sums to 0.

� Let m ∈ R have the smallest absolute value such thatf + mg = 0 on an edge of K . Then f + mg is a matching withmore 0 edges than f , a contradiction.

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Matchings

Theorems

Claim 3 (continued)

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Matchings

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Claim 3 (continued)

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Matchings

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Summary of Claims

� Therefore, H consists only of components that are odd cycles orcopies of K2.

� On copies of K2, f = 1, and on odd cycles, f = 1/2. Sincef = 0 on the rest of G , f only takes on values in {0, 1/2, 1}.

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Matchings

Theorems

Summary of Claims

� Therefore, H consists only of components that are odd cycles orcopies of K2.

� On copies of K2, f = 1, and on odd cycles, f = 1/2. Sincef = 0 on the rest of G , f only takes on values in {0, 1/2, 1}.

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Matchings

Theorems

Fractional Transversals

� A fractional transversal of G is a function g : V (G )→ [0, 1]such that

∑v∈e g(v) ≥ 1 for every e ∈ E (G ).

� The fractional transversal number is the infimum of∑v∈V (G) g(v) over all fractional transversals g . By duality, this

is α′∗(G ).

� Theorem 2.1.6: For every graph G , there exists a fractionaltransversal g for which∑

v∈V (G)

g(v) = α′∗(G )

such that g(v) ∈ {0, 1/2, 1} for every vertex v .

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Matchings

Theorems

is α′∗(G ).

v∈V (G)

g(v) = α′∗(G )

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Matchings

Theorems

is α′∗(G ).

v∈V (G)

g(v) = α′∗(G )

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Matchings

Theorems

Perfect Matchings

� A perfect matching is a set of edges such that every vertex is insome edge of the matching.

� A fractional perfect matching is a fractional matching f suchthat

∑e f (e) = |V (G )|/2.

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Matchings

Theorems

Perfect Matchings

� A perfect matching is a set of edges such that every vertex is insome edge of the matching.

� A fractional perfect matching is a fractional matching f suchthat

∑e f (e) = |V (G )|/2.

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Matchings

Theorems

Tutte’s Theorem

� Let o(G ) be the number of odd components and i(G ) thenumber of isolated vertices.

� Tutte’s Theorem: A graph G has a perfect matching if andonly if o(G − S) ≤ |S | for all S ⊆ V (G ).

� Fractional Tutte’s Theorem: A graph G has a fractionalperfect matching if and only if i(G − S) ≤ |S | for all S ⊆ V (G ).

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Matchings

Theorems

Tutte’s Theorem

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Matchings

Theorems

Tutte’s Theorem

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Matchings

Theorems

Bounds

Consider some quick bounds on χ′(G ).

� χ′(G ) ≥ ∆(G )

� χ′(G ) ≥

⌈e(G )

bv(G )/2c

Proof.

Each color class is a matching, and matchings have at mostbv(G )/2c edges. Apply the pigeonhole principle.

� We can improve this slightly to get χ′(G ) ≥ maxH

⌈e(H)

bv(H)/2c

where H is a subgraph of G . This is because χ′(G ) ≥ χ′(H).

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Matchings

Theorems

Bounds

� χ′(G ) ≥ ∆(G )

� χ′(G ) ≥

⌈e(G )

bv(G )/2c

Proof.

⌈e(H)

bv(H)/2c

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Matchings

Theorems

Bounds

� χ′(G ) ≥ ∆(G )

� χ′(G ) ≥

⌈e(G )

bv(G )/2c

Proof.

⌈e(H)

bv(H)/2c

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Matchings

Theorems

Bounds

� χ′(G ) ≥ ∆(G )

� χ′(G ) ≥

⌈e(G )

bv(G )/2c

Proof.

⌈e(H)

bv(H)/2c

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Matchings

Theorems

Bounds

In searching for a subgraph H that achieves the maximum, we may

assume that |V (H)| is odd. To see why, suppose maxH

⌈e(H)

bv(H)/2c

⌉is achieved by H where |V (H)| is even and let v ∈ V (H) haveminimum degree.

δ(H) ≤ 2|E (H)||V (H)|

|E (H)| − δ(H)12 (|V (H)| − 2)

≥ 2|E (H)||V (H)|

|E (H − v)|b|V (H − v)|/2c

≥ |E (H)|b|V (H)|/2c

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Matchings

Theorems

Bounds

In searching for a subgraph H that achieves the maximum, we may

assume that |V (H)| is odd. To see why, suppose maxH

⌈e(H)

bv(H)/2c

⌉is achieved by H where |V (H)| is even and let v ∈ V (H) haveminimum degree. Then

δ(H) ≤ 2|E (H)||V (H)|

|E (H)| − δ(H)12 (|V (H)| − 2)

≥ 2|E (H)||V (H)|

|E (H − v)|b|V (H − v)|/2c

≥ |E (H)|b|V (H)|/2c

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Matchings

Theorems

Bounds

� Therefore, we may assume H is induced and has an odd numberof vertices. We may also suppose there are at least threevertices.

� Define

Λ(G ) = maxH

2|E (H)||V (H)| − 1

where H is as stated previously. (For |V (G )| < 3, let Λ(G ) = 0.)

� Then χ′(G ) ≥ max{∆(G ), dΛ(G )e}. The same holds for χ′t/t,and thus χ′f (G ) ≥ max{∆(G ),Λ(G )}.

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Matchings

Theorems

Bounds

� Define

Λ(G ) = maxH

2|E (H)||V (H)| − 1

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Matchings

Theorems

Bounds

� Define

Λ(G ) = maxH

2|E (H)||V (H)| − 1

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Matchings

Theorems

Fractional Chromatic Number

In fact,χ′f (G ) = max{∆(G ),Λ(G )}.

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Matchings

Theorems

Vizing’s Theorem

� Vizing’s Theorem: For any graph G ,∆(G ) ≤ χ′(G ) ≤ ∆(G ) + 1.

� Fractional Vizing’s Theorem: For any graph G ,∆(G ) ≤ χ′f (G ) ≤ ∆(G ) + 1.

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Matchings

Theorems

Vizing’s Theorem

� Vizing’s Theorem: For any graph G ,∆(G ) ≤ χ′(G ) ≤ ∆(G ) + 1.

� Fractional Vizing’s Theorem: For any graph G ,∆(G ) ≤ χ′f (G ) ≤ ∆(G ) + 1.

fractional edge coloring

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