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Edge colorings
Edge coloring problems often arise when objects being scheduled are pairsof underlying elements, e.g., a pair of teams that play each other, a pairconsisting of a teacher and a class, etc. (See slides for an example.)
Definition A k-edge-coloring of a graph G is a functionf : E(G) ! {1, 2, . . . , k}. It is proper if edges that share an endpoint havedi↵erent colors, i.e.,
e1 \ e2 6= ; =) f (e1) 6= f (e2).
G is said to be k-edge-colorable if it has a proper k-edge-coloring.
Definition The minimum k such that G has a proper k-edge-coloring iscalled the edge chromatic number of G and is denoted �0(G).
Remark Graphs with loops cannot be properly colored, so we considerloopless graphs only.
Question What are easy upper and lower bounds on �0(G)?
Proposition. For any graph G, �(G) �0(G) |E(G)|.
The above lower bound may seem trivial. However, this bound holds withequality for large classes of graphs – in particular, for bipartite graphs,which we will prove .
Examples
• For a cycle Cn, �(Cn) = 2 and
�0(Cn) =
(2 if n even
3 if n odd.
Math 104 - Prof. Kindred - Lecture 15 Page 1
• The complete graph K2n has �(K2n) = 2n�1. Then we can color theedges of K8 as illustrated below:
Exercise outside of class Show that �0(K2n+1) = 2n + 1.
Remark Note that in vertex coloring, each color class was an independentset. In the context of edge coloring, each color class is a matching.
Recall that �(G) � n↵(G) since color classes are independent sets for vertex
coloring. Can we obtain an analogous result for �0(G)?
Proposition. For any graph G, �0(G) � |E(G)|↵0(G) .
Fundamental theorem coming up =) �0(G) is always very close to �(G)for simple graphs G.
Theorem (Vizing). Let G be a simple graph (no loops, no multiple
edges). Then
�(G) �0(G) �(G) + 1.
We omit the proof of the upper bound, but you can find it in the textbook(algorithmic proof).
Remark The simple graphs G for which �0(G) = �(G) are called Class
1 graphs, and those for which �0(G) = �(G)+1 are called Class 2 graphs.
The problem of deciding to which class a given graph belongs is very hard(NP-complete).
Math 104 - Prof. Kindred - Lecture 15 Page 2
Question Does this result hold for graphs with multiple edges (looplessgraphs)?
Counterexample: “fat” triangle
Given 3 vtcs, say there are l edges between each pair of vtcs. (So the graphhas 3l edges total.) Then �(G) = 2l and �0(G) = 3l.
The generalization of Vizing’s theorem for loopless graphs (or multigraphswithout loops) is the following.
Theorem. For any (loopless) graph G,
�(G) �0(G) �(G) + µ(G)
where µ(G) = maxuv2E(G) µ(u, v) and µ(u, v) is the # of edges with
endpoints u and v.
So µ(G) is the maximum of the edge multiplicities in G.
Our next goal is to show that bipartite graphs are Class 1 graphs – that is,�(G)-edge-colorable. We first need a lemma.
Lemma. Every bipartite graph G has a �(G)-regular bipartite super-
graph (a graph containing G).
Proof. LetG be anX, Y -bipartite graph with k = �(G). A large k-regularsupergraph G0 of G can be constructed as follows.
• Clone G. If G is not regular, add a vertex to X for each vertex of Yand a vertex to Y for each vertex of X . On the new vertices, constructanother copy of G.
Math 104 - Prof. Kindred - Lecture 15 Page 3
• Add edges between vertex and its clone, as needed. Foreach v 2 V (G) with dG(v) < k, join its two copies in the new graphwith k � dG(v) edges to get G0.
Now G0 is a k-regular bipartite supergraph of G. (G0 is connected if G wasconnected.) ⌅
See slides for an example of above construction.
Remark IfG is a simple bipartite graph, then we can use a variation of theconstruction above to obtain a simple �(G)-regular bipartite supergraphof G.
Replace 2nd step above with the following: For each v 2 V (G)with dG(v) < k, join its two copies with a single edge in the newgraph to get G0.
Then�(G0) = �(G) = k,
�(G0) = �(G) + 1,
and G0 is a supergraph of G. Iterating the 2-step process k� �(G)times yields the desired simple bipartite supergraph H .
Recall Hall’s Marriage Thm:
Theorem. For k 2 N, every k-regular bipartite graph has a
perfect matching.
Math 104 - Prof. Kindred - Lecture 15 Page 4
Theorem (Konig). If G is a bipartite graph, then �0(G) = �(G).
Proof. Since �0(G) � �(G) for any graph G, it is su�cient to show that�0(G) �(G) when G is bipartite.
We give a proof by induction on �(G).
Base case: Assume �(G) = 1. Then no two edges share an endpoint(since such a shared vertex would have degree at least 2), and so every edgeof G can be assigned the same color. Thus, �0(G) 1 = �(G).
Induction hypothesis: Suppose any bipartite graphG with�(G) = k
has �0(G) �(G).
Suppose G is a bipartite graph with �(G) = k + 1. By previous lemma,there exists a �(G)-regular bipartite graph H that contains G. By Hall’sMarriage Thm, H has a perfect matching M . Then H �M is a k-regularbipartite graph and so by the IH, H �M satisfies
�0(H �M) �(H �M) = �(H)� 1 = k.
Consider any k-edge-coloring of H �M . Extend it to an edge-coloring ofH by assigning all edges of M color k + 1. This is a proper (k + 1)-edge-coloring of H since no two edges of M share an endpoint.
Since any subgraph of H must then be (k+1)-edge-colorable, we have that�0(G) k + 1. ⌅
Hmwk problem 7.1.21 asks you to give an algorithmic proof of Konig’sThm. The proof yields a polynomial-time algorithm for constructing a�(G)-edge-coloring of a bipartite graph G.
Math 104 - Prof. Kindred - Lecture 15 Page 5
Line graphs
Definition Given a graph G, the line graph of G, denoted L(G), is thegraph whose vertices are the edges of G and
ef 2 E(L(G)) () e and f are edges in G with a common endpoint.
A line graph is the intersection graph of the edges of G.
Example
G L(G)G L(G)
Observations
• If e = uv is an edge in G, then dL(G)(e) = dG(u) + dG(v)� 2.
• The numbers of vertices and edges in L(G) are
|V (L(G))| = |E(G)| and |E(L(G))| =X
v2V (G)
✓dG(v)
2
◆
• The line graph of a connected graph G is connected. (Note that theconverse is not necessarily true: a disconnected graph may have aconnected line graph.)
• For a simple graph G, vertices form a clique in L(G) if and only if thecorresponding edges in G have one common endpoint (a star) or forma triangle. (Thus, !(L(G)) = �(G) unless �(G) = 2 and G containsa triangle.)
Math 104 - Prof. Kindred - Lecture 15 Page 6
• The only connected graph that is isomorphic to its line graph is a cyclegraph Cn for n � 3.
• For a graph G, �0(G) = �(L(G)). So edge coloring is really a specialcase of vertex coloring.
• We can characterize the graphs that exist as line graphs. A graph H isthe line graph of some other graph if and only if it is possible to find acollection of cliques in H , partitioning the edges of H , such that eachvertex of H belongs to at most two of the cliques.
• A forbidden subgraph characterization exists for line graphs, namely aset S of nine graphs exists such that a simple graph G is a line graphof some simple graph if and only if G does not have any graph in S asan induced subgraph.
This characterization yields an algorithm to test whether or not G isa line graph in polynomial time (in n = |V (G)|).
Math 104 - Prof. Kindred - Lecture 15 Page 7
Edge coloring and line graphs
Math 104, Graph TheoryMarch 14, 2013
Scheduling problemEdge colorings
Edge coloring problems often arise when objects being scheduled are pairsof underlying elements, e.g., a pair of teams that play each other, a pairconsisting of a teacher and a class, etc.
Problem
In a school, there are a teachers x1, x2, . . . , xa and b
classes y1, y2, . . . , yb. Given that teacher xi is requiredto teach class yj for pij periods, schedule a completetimetable in the minimum # of possible periods.
Define a bipartite graph with
• a set X of a vtcs representing teachers,
• a set Y of b vtcs representing classes
• and there are pij edges between xi 2 X and yj 2 Y .
The least # of periods needed to schedule all classes is the least # of colorsneeded to properly edge-color the graph.
Definition A k-edge-coloring of a graph G is a functionf : E(G) ! {1, 2, . . . , k}. It is proper if edges that share an endpoint havedi↵erent colors, i.e.,
e1 \ e2 6= ; =) f (e1) 6= f (e2).
G is said to be k-edge-colorable if it has a proper k-edge-coloring.
Definition The minimum k such that G has a proper k-edge-coloring iscalled the edge chromatic number of G and is denoted �0(G).
Math 104 - Prof. Kindred - Lecture 15 Page 1
Edge-coloring a complete graph
Building a bipartite, regular supergraph
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Forbidden subgraph characterization for line graphs