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Optimal Parity Edge-Coloring of Complete Graphs David P. Bunde∗, Kevin Milans†, Douglas B. West‡, Hehui Wu§ A parity walk in an edge-coloring of a graph is a walk along which each color is used an even number of times. Let p(G) be the least number of colors in an edge-coloring of G having no parity path (a parity edge-coloring). Let p(G) be the least number of colors in an edge-coloring of G having no open parity walk (a strong parity edge-coloring).
Always p(G) ≥ p(G) ≥ χ (G). We prove that p(Kn) = 2 lg n − 1 for all n. The optimalstrong parity edge-coloring of Kn is unique when n is a power of 2, and the optimalcolorings are completely described for all n.
Our work began by studying which graphs embed in the hypercube Qk, the graph with vertexset {0, 1}k in which vertices are adjacent when they differ in exactly one coordinate. Color- ing each edge with the position of the bit in which its endpoints differ yields two necessary conditions for the coloring inherited by a subgraph G: 1) every cycle uses each color an even number of times, 2) every path uses some color an odd number of times.
Existence of a k-edge-coloring satisfying conditions (1) and (2) is also sufficient for a con- nected graph G to be a subgraph of Qk. This characterization of subgraphs of Qk was provedas early as 1972, by Havel and Mov´ arek [6]. The problem was studied as early as 1953 by Define the usage of a color on a walk to be the parity of the number of times it appears as the walk is traversed. A parity walk is a walk in which the usage of every color is even.
Condition (1) for an edge-coloring states that every cycle is a parity walk, and (2) states ∗Computer Science Department, Knox College, Galesburg, IL, Partially supported †Department of Computer Science, University of Illinois, Urbana IL,‡Department of Mathematics, University of Illinois, Urbana, IL, Work supported in part by the NSA under Award No. MDA904-03-1-0037.
§Department of Mathematics, University of Illinois, Urbana, IL.
In general, a parity edge-coloring is an edge-coloring with no parity path, and a strong parity edge-coloring (spec) is an edge-coloring with no open parity walk (that is, every parity walk is closed). Although some graphs do not embed in any hypercube, using distinct colors on the edges produces a spec for any graph. Hence the parity edge-chromatic number p(G) and the strong parity edge-chromatic number p(G), defined respectively to be the minimum numbers of colors in a parity edge-coloring of G and a spec of G, are well defined. Elementary results on these parameters appear in [4].
When T is a tree, p(T ) = p(T ) = k, where k is the least integer such that T embeds in Qk. Since incident edges of the same color would form a parity path of length 2, everyparity edge-coloring is a proper edge-coloring, and hence p(G) ≥ χ (G), where χ (G) denotes the edge-chromatic number. Although there are graphs G with p(G) > p(G) [4], it remains unknown how large p(G) can be when p(G) = k. It also remains unknown whether there is a bipartite graph G with p(G) > p(G).
When n is a power of 2, we will prove that the optimal spec of Kn is unique, which will help us determine p(Kn) for all n. With a suitable naming of the vertices, we call thisedge-coloring of Kn the “canonical” coloring.
F , let K(A) be the complete graph with vertex set A.
canonical coloring of K(A) is the edge-coloring f defined by f (uv) = u + v, where u + v denotes binary vector addition. When n = 2k, we treat the vertex as F , the canonical coloring of K(A) is a spec. Consequently, if n = 2k, then p(Kn) = p(Kn) = χ (Kn) = n − 1.
Proof. If W is an open walk, then its endpoints differ in some bit i. Thus in the canonical coloring the total usage of colors flipping bit i along W is odd, and hence some color has odd usage on W . The canonical coloring of K( k F ) uses 2k − 1 colors (the color 0k is not used).
Since every complete graph is a subgraph of the next larger complete graph, we obtain p(Kn) ≤ 2 lg n − 1. In Section 2, we use linear algebra to show that this upper bound isexact; this is our main result.
The canonical coloring is relevant to a less restrictive edge-coloring problem. A walk of length 2k is repetitive if, for each 1 ≤ i ≤ k, the ith and (k + i)th edges have the same color.
A Thue coloring is an edge-coloring having no repetitive path, and the Thue number t(G) is the minimum number of colors in a Thue coloring of G. Every parity edge-coloring is a Thue coloring, and Alon, Grytczuk, Haluszczak, and Riordan [2] showed that t(Kn) ≤ 2 lg n − 1using the canonical coloring. Obtaining non-trivial lower bounds on t(Kn) remains an openproblem. Our result implies that a Thue coloring of Kn better than the canonical coloringmust contain a parity walk.
To further motivate our focus on complete graphs, we note that our main result implies a special case of Yuzvinsky’s Theorem [10]. Yuzvinsky’s Theorem provides a tight lower bound on the number of distinct sums induced by two sets of binary vectors and makes use of the Hopf–Stiefel–Pfister function.
Definition 1.3 (Hopf–Stiefel–Pfister function) If r and s are positive integers, define r ◦ s to be the least integer n such that (x + y)n is in the ideal (xr, ys) of F2[x, y].
Hence, r ◦ s is the least n such that for each k with n − s < k < r, n is even. Recently, Plagne [8] found a simple formula for the Hopf–Stiefel–Pfister function, and K´ Theorem 1.4 ([8]; see also [7]) r ◦ s = mink∈ Theorem 1.5 (Yuzvinsky [10]; see also [1], [3], [5]) Given A, B ⊆ A, b ∈ B}. If |A| ≥ r and |B| ≥ s, then |C| ≥ r ◦ s.
Our main result implies the special case of Yuzvinsky’s Theorem where A = B. Indeed, if F and |A| = r, then the canonical (and optimal) coloring of K(A) uses 2 lg r − 1 colors, none of which is 0k. By construction, these colors all lie in C; therefore |C| ≥ 2 lg r = r ◦ r.
The canonical coloring extends to complete bipartite graphs in a natural way: if A, B ⊆ F and K(A, B) is the complete bipartite graph with partite sets A and B, then the edge- coloring defined by f (ab) = a + b is a spec. Because Yuzvinsky’s Theorem is tight, for each F with |A| = r, |B| = s, and |C| = r◦s. Consequently, p(K We conjecture that equality holds. A direct proof in the graph-theoretic setting would imply In this section, we use linear algebra to prove that p(Kn) ≥ 2 lg n − 1. The main idea isto introduce an additional vertex without needing additional colors until a power of 2 is reached. We begin by proving that every optimal spec of Kn is a canonical coloring when nis a power of 2.
Definition 2.1 An edge-coloring f of a graph G satisfies the 4-constraint if whenever f (uv) = f (xy) and vx ∈ E(G), also uy ∈ E(G) and f (uy) = f (vx).
Lemma 2.2 If f is a parity edge-coloring in which every color class is a perfect matching, Proof. Otherwise, given f (uv) = f (xy), the edge of color f (vx) incident to u forms a parity path of length 4 with uv, vx, and xy.
Theorem 2.3 If f is a parity edge-coloring of Kn in which every color class is a perfectmatching, then f is a canonical coloring and n is a power of 2.
Proof. Every edge is a canonically colored copy of K2. Let R be a largest vertex set suchthat |R| is a power of 2 and f restricts to a canonical coloring on R. Define j by |R| = 2j−1.
under which f is the canonical coloring.
Since f is canonical, every color used within R by f forms a matching of R. Let c be a color not used within R; since c is used on a perfect matching, c matches R to some set U .
F as follows: for x ∈ R, obtain φ (x) by appending 0 to φ(x); for x ∈ U obtain φ (x) by appending 1 to φ(x ), where x is the neighbor of x in color c. Within R , we henceforth refer to the vertices by their names under φ .
By Lemma 2.2, the 4-constraint holds for f . The 4-constraint copies the coloring from the edges within R to the edges within U . To see this, consider x , y ∈ U arising from x, y ∈ R, with f (xx ) = f (yy ) = c. Now f (x y ) = f (xy) = x + y = x + y , using the 4-constraint, the fact that f is canonical on R, and the definition of φ . Hence f is canonical Finally, let u be the name of the color on the edge 0ju, for u ∈ U . For any v ∈ R, let w = u + v; note that w ∈ U . Both 0jv and uw have color v, since f is canonical within R and within U . By applying the 4-constraint to {v0j, 0jw, wu}, we conclude that f (uv) = f (0jw) = w. Since w = u + v, this completes the proof that f is canonical on R .
In connection with this uniqueness result, Mubayi asked whether a stability property holds. That is, when n is a power of 2, does there exist a parity edge-coloring or a spec of Kn that has only (1 + o(1))n colors but is “far” from the canonical coloring? Now we begin the algebraic observations needed for the main result. Relative to any k-edge-coloring f , the parity vector π(W ) of a walk W is the binary k-tuple whose ith bit agrees in parity with the usage of color i along W . Let the parity space Lf be the set ofparity vectors of closed walks. We note that Lf is a vector space.
Lemma 2.4 If f is an edge-coloring of a connected graph G, then Lf is a binary vectorspace.
f ⊆ F , it suffices to show that L is closed under binary addition. Given a u, u-walk W and a v, v-walk W , let P be a u, v-path in G, and let P be its reverse. Following W, P, W , P in succession yields a u, u-walk with parity vector π(W ) + π(W ).
For a binary vector space L, let w(L) denote the least number of nonzero coordinates of any vector in L. For an edge-coloring f of Kn, w(Lf ) determines whether f is a spec.
Lemma 2.5 If an edge-coloring f of a graph G is a spec, then w(Lf ) ≥ 2. The converseholds when G = Kn.
Proof. If the parity vector of a closed walk W has weight 1, then one color has odd usage in W (say on edge e). Now W − e is an open parity walk, and f is not a spec.
If f is not a spec, then π(W ) = 0 for some open walk W . In Kn, the ends of W are adjacent, and adding that edge yields a closed walk whose parity vector has weight 1.
Lemma 2.6 For colors a and b in an optimal spec f of Kn, there is some closed walk Won which the colors having odd usage are a, b, and one other.
Proof. We use Lemma 2.5 repeatedly. Since f is optimal, merging the colors a and b into a single color a yields an edge-coloring f that is not a spec. Hence under f there is a closed walk W on which f has odd usage for only one color c. Also c = a , since otherwise f has odd usage on W for only a or b. With c = a and the fact that f has odd usage for at least two colors on W , both a and b also have odd usage on W , and W is the desired walk.
The same idea as in Lemma 2.6 shows that w(Lf ) ≥ 3 when f is an optimal spec of Kn, but we do not need this observation. We note, however, that the condition w(Lf ) ≥ 3 isthe condition for Lf to be the set of codewords for a 1-error-correcting code. Indeed, whenn = 2k and f is the canonical coloring, Lf is a perfect 1-error-correcting code of length n − 1.
A dominating vertex in a graph is a vertex adjacent to all others.
Lemma 2.7 If f is an edge-coloring of a graph G with a dominating vertex v, then Lf isthe span of the parity vectors of triangles containing v.
Proof. By definition, the span is contained in Lf . Conversely, consider any π(W ) ∈ Lf .
Let S be the set of edges with odd usage in W , and let H be the spanning subgraph of G with edge set S. Since the total usage at each vertex of W is even, H is an even subgraph of G. Hence H decomposes into cycles, which are closed walks, and π(W ) is the sum of the It therefore suffices to show that S is the set of edges that appear in an odd number of the triangles formed by v with edges of H − v. Each edge of H − v is in one such triangle, so we need consider only edges involving v. An edge vw lies in an odd number of these triangles if and only if dH−v(w) is odd, which occurs if and only if w ∈ NH(v), since dH(w) is even.
By definition, vw ∈ E(H) if and only if vw has odd usage in W and hence lies in S.
Lemma 2.8 If f is an optimal spec of Kn that uses some color less than n/2 times, then fextends to a spec of Kn+1 using the same colors.
Proof. View Kn+1 as arising from Kn by adding a vertex u. Let a be a color used less thann/2 times by f , and let v be a vertex of Kn at which a does not appear.
We use f to define f on E(Kn+1). Let f agree with f on E(Kn), and let f (uv) = a. To define f on each remaining edge uw, first let b = f (vw). By Lemma 2.6, there is a closed walk W with odd usage precisely for a and b and some third color c under f . Let f (uw) = c.
Note that f uses the same colors as f . It remains only to show that f is a spec. To do this we prove that w(Lf ) ≥ 2, by showing that Lf ⊆ Lf . By Lemma 2.7, it suffices to showthat π(T ) ∈ Lf when T is a triangle in Kn+1 containing v.
Triangles not containing u lie in the original graph and have parity vectors in Lf . Hence we consider the triangle T formed by {u, v, w}. Now π(T ) = π(W ) ∈ Lf , where W is thewalk used to specify f (uw).
Proof. If some color class in an optimal spec is not a perfect matching, then p(Kn) =p(Kn+1), by Lemma 2.8. This vertex absorption cannot stop before the number of verticesreaches a power of 2, because when every color class is a perfect matching the coloring is canonical, by Theorem 2.3. It cannot continue past 2 lg n vertices, since then the maximum degree equals the number of colors. Hence p(Kn) = p(K2 lg n ) = 2 lg n − 1.
Corollary 2.10 If f is an optimal spec of Kn, then f is obtained by deleting vertices fromthe canonical coloring of K2 lg n .
Proof. By Lemma 2.8, we may extend f to an optimal spec f of K2 lg n ; by Theorem 2.3,f is the canonical coloring.
It is natural to wonder whether every edge-coloring of Kn that satisfies the 4-constraint is a spec or a parity edge-coloring. One can construct examples that show the answer is no.
Similarly, not every parity edge-coloring of Kn is a spec. Nevertheless, it may be that everyoptimal parity edge-coloring is a spec. We offer the following conjecture, which in [4] we Conjecture 2.11 p(Kn) = p(Kn) for every positive integer n.
We thank Dan Cranston, Will Kinnersley, Brighten Godfrey, Michael Barrus, and Mohit [1] N. Alon, Combinatorial Nullstellensatz. Combin., Prob., and Computing 8 (1999), 7-29.
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[4] D. P. Bunde, K. Milans, D. B. West, and H. Wu, Parity Edge-Coloring of Graphs. preprint.
[5] S. Eliahou and M. Kervaire, Sumsets in vector spaces over finite fields. J. Number Theory 71 arek, B-valuation of graphs. Czech. Math. J. 22, 338–351.
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