A VARIANT OF RECONSTRUCTIBILITY OF COLORED GRAPHS

A variant of reconstructibility of colored graphs is defined, and some facts proved. Some computations facts from an earlier paper are revised. AMS Subj. Classification: 05C60


Introduction
Colored graph reconstructibility has been considered since the early 1970's (see [1]).More recent references include [7], [6], [3].Using terminology from [3], define a C-graph to be a graph, with colors assigned to the vertices and edges.C-graphs are called colored graphs in the literature.An isomorphism φ of C-graphs must preserve colors (i.e., for a vertex v φ(v) must have the same color as v and similarly for edges).A C-graph is defined to be reconstructible if it is determined by its deck.That is, if G and H have the same deck, in that the members of the two decks can be paired as isomorphic pairs, then G and H are isomorphic.
Given C-graphs G and H, define φ to be a C-isomorphism if φ(v 1 ) and φ(v 2 ) have the same color whenever v 1 and v 2 do; and similarly for edges.A C-graph is defined to be C-reconstructible if, whenever G and H have C-isomorphic decks, in that the members of the two decks can be paired as C-isomorphic pairs, then G and H are C-isomorphic.
Sections 6 and 7 of [3] contain many errors due to confusion of C-reconstructibility C-reconstructibility, and will be completely revised here.
All graphs will be assumed to have at least three vertices.For a graph G V (G) denotes the vertices, E(G) the edges, and for v ∈ V (G) G v denotes the point-deleted subgraph.

Basic facts
Define a C-graph G to be a graph, together with partitions of its set of vertices and set of edges.An isomorphism between C-graphs must preserve the partitions.If v is a vertex, G v is the point deleted subgraph, together with the induced partitions, where two vertices or edges belong to the same part in G v iff they do in G.To a Cgraph G there corresponds a C-graph Ḡ, where a part is the vertices or edges of a given color.
Theorem 1. Two C-graphs G, H are C-isomorphic iff Ḡ, H are isomorphic.
Proof.Indeed, a bijection φ from the vertex set Proof.Suppose Ḡ is reconstructible and G, H have C-isomorphic decks.By theorem 1 Ḡ, H have isomorphic decks, whence by hypothesis Ḡ, H are isomorphic, whence by theorem 1 G, H are C-isomorphic.Suppose G is C-reconstructible and Ḡ, H have isomorphic decks.A similar argument shows that Ḡ, H are isomorphic.
As in [3] define a V-graph to be a graph, with colors assigned to the vertices (alternatively a C-graph with constant edge color); and an E-graph to be a graph with edge colors.Similarly a V-graph (resp.Ē-graph) is a graph with a vertex (resp.edge) partition.
The notion of Ē-reconstructibility is of little interest.Indeed, all three edge partitionings of K 3 have the same deck.There are 25 edge partitionings of K 4 , having 11 decks.Hereafter, only V-graphs will be considered.
Theorem 3. The multiset of part sizes of a V-graph is reconstructible.
Proof.Letting G denote the graph and n v the number of vertices, the part size multiset of G is 1 nv iff the part size multiset of each G v is 1 nv−1 .Otherwise, the number of parts is the maximum such among the G v .Let S be the lexicographically greatest part size multiset among the G v ; the part size multiset of G is readily obtained from S.
Proof.This value is the largest size of a part of G, whose multiplicity is 1 less in G v .
A basic fact about V-graphs is that a disconnected V-graph is reconstructible.Essentially the same argument (see theorem 3 of [3]) shows that for a V-graph G, the components together with their vertex partitions are reconstructible.However, it does not follow (at least readily) that G is reconstructible.
If G is a V-graph, G may be represented by a bipartite graph G r which has a vertex class V for the vertices of G and a vertex class C for the colors.The edges of G r are those of G, an an edge {v, c} if v has color c.It is readily seen that given two V-graphs G, H with the same colors, G is isomorphic to H iff G g and H r are isomorphic by an isomorphism fixing V setwise and C pointwise; and Ḡ is isomorphic to H iff G g and H r are isomorphic by an isomorphism fixing V and C setwise.This observation will be used in the computations below.

Computations for V-graphs
This section revises section 6 of [3].
Proof.For |V (G)| = 3 the 14 cases of Ḡ may be enumerated, and the decks seen to be distinct.
For |V (G)| ≥ 4 the claim may be verified by a computer program.By results of [4] the underlying graph G is reconstructible.By theorem 6 only G where |E(G)| ≤ n(n − 1)/4 need be considered.By theorem 3, letting P denote the multiset of vertex partition part sizes, the V-graphs for each G and P may be considered separately.Representing them as noted above, the V-graphs may be canonicalized up to setwise fixing of the partition parts using the Nauty [5] library.Reconstructibility may be verified by canonicalizing the decks, and verifying that distinct canonicalized V-graphs have distinct canonicalized decks.
Proof.By theorem 7, the V-graphs with a given V-graph may be considered separately.In a vertex coloring, two parts may not have their colors exchanged if (A) they have different sizes, or (B) they have different degree sequences.
For |V (G)| = 3, for 6 of 14 V-graphs there is a single isomorphism class of vertex colorings, for 6 of them there are two classes which may be distinguished by criterion (A), and for 2 of them there are three classes which may be distinguished by criterion (B).
For |V (G)| ≥ 4 the claim may be verified by a computer program.The V-graphs may be canonicalized "on the fly", one graph at a time.By standard results on V-reconstructibility (see [3]), only G need be considered, which are connected, have at most half the possible edges present, and are not regular.For each V-graph, the V-graphs may be generated and canonicalized.The parts may be grouped, where in a group the size and degree sequence is the same.
Each group is assigned a distinct set of colors, and colors assigned to the nodes of a part in all possible ways.Algorithm 2.14 of [2] is useful in this step.A check is made that the decks are distinct.

Computations for E-graphs
This section revises section 7 of [3].The claims will be stated as theorems; they have already appeared in [7], More detailed proofs will be given here.Recall from [3] that a graph G is said to be Ereconstructible if every edge coloring of G is reconstructible.Recall also that the multiset of colored edges is reconstructible, whence the multiset incident to the vertex v is known for G v .From hereon let G denote an edge coloring of K n .
Proof.G is reconstructible from any G v by adding the other two edges.
Proof.The proof may be divided into cases.Case T1, there is a monochromatic triangle.The remaining edges may be added arbitrarily.
Case S1, there is a monochromatic star.The remaining edges may be added arbitrarily.
Case T3, there is a 3 colored triangle.Let 123 be the colors and xyz the colors of the other 3 edges, the complementary star.The other 3 stars are colored 12x, 13y, and 23z.If these are distinct sets then G is readily reconstructed.Otherwise, w.l.g.x=3 and y=2.Whether or not z=1 G is readily reconstructed.
Case S3, there is a 3 colored star.This is similar to case T3, with stars and triangles interchanged.
In the remaining case, there is a 112 star and an xyz triangle, where in the other 3 triangles 12x, 12y, 11z, x and y are 1 or 2 and z is 2 or 3.Both the cases z=3 and z=2 are readily reconstructible.
Proof.Let P be a partition of n e , the number of edges.Assign n i colors to part i, where n i is the value of part i.Let G be K n with a partition Q of the edges, with part size list P .Let S P be the set of canonicalized such G (writing a file of these may be done first).For G ∈ S P with set partition Q let T Q be the set of edge colorings of K n which agree with the colors assigned to P .It suffices to verify by computer that for each P , the graphs in ∪ Q T Q have distinct decks.
As a preliminary step, the number partitions 1 10 , 21 8 , 31 7 , and 2 2 1 6 may be omitted, since G may be seen to be reconstructible in these cases.Indeed, there is a vertex v such that in G v the edge colors are distinct and there is an edge incident to v whose color is not one of these.G may be reconstructed from G w where w is a vertex other than v.
As noted in [3] even enumerating the set partitions of a 15 element set requires a fairly extensive computation.Further discussion of K 6 is omitted.