Decomposition of complete graphs into connected unicyclic bipartite graphs with eight edges

We prove that each of the 34 non-isomorphic connected unicyclic bipartite graphs with eight edges decomposes the complete graph Kn whenever the necessary conditions are satisfied.


Introduction
Let U be a graph on m vertices. A decomposition of the graph U is a family of pairwise edge disjoint subgraphs D = {G 0 , G 1 , . . . , G s } such that every edge of U belongs to exactly one member of D. If each subgraph G i is isomorphic to a given graph G we speak about G-decomposition of U (or we say that G forms a decomposition of U ), or a G-design when U ⇠ = K m . The decomposition of K m is cyclic if there exists an ordering (x 1 , x 2 , . . . , x m ) of the vertices of K m and isomorphisms i : G 0 ! G i , i = 0, 1, 2, . . . , n 1, such that i (x j ) = x i+j for each j = 1, 2, . . . , m. Subscripts are taken modulo m.
A graph G is unicyclic if it contains exactly one cycle. In this paper, we provide the necessary and sufficient conditions for decompositions of complete graphs into each of the 34 connected unicyclic bipartite graphs with eight edges.
Kang, Yuan, and Liu researched graphs with six vertices and eight edges in 2005 [7]. There are 22 non-isomorphic graphs of this type, and they proved the following theorem with respect to decompositions of complete graphs. Theorem 2.6 (Kang, Yuan, Liu 2005). Let G be a connected graph with six vertices and eight edges. There exists a G-decomposition of K n if and only if n ⌘ 0, 1 (mod 16) and n 16 with two possible exceptions for n = 32.

Definitions and tools
First we define the labelings that will be used in our work. We start with a ⇢-labeling, first defined by Rosa [9]. Note that Rosa originally used the term valuation instead of labeling.
Definition 3.1. Let G be a graph with n edges. A ⇢-labeling of G is a one-to-one function f : A more restrictive version is the -labeling, also introduced by Rosa in [9].
An even more restrictive labeling is the well-known graceful labeling (which was originally called a -valuation by Rosa [9]). Definition 3.3. Let G be a graph with n edges. A graceful labeling is a -labeling f with the additional restriction that f : V (G) ! {0, 1, . . . , n}, that is, the labels are assigned integers from the set {0, 1, . . . , n} instead of {0, 1, . . . , 2n}.
Rosa [9] proved that if a graph G with n edges has one of the above labelings, then a decomposition of the complete graph K 2n+1 exists.  5. An ↵-labeling is a graceful labeling with the additional property that there exists an integer such that for each edge xy either f (x)  < f(y) or f (y)  < f(x).
While the ↵-labeling (defined also by Rosa in [9]) is a more restrictive form of graceful labeling, it facilitates decompositions of a much larger set of complete graphs, that is, decompositions of K 2nk+1 instead of just K 2n+1 , where k is an arbitrary positive integer. An ↵-labeled graph must be bipartite, and when V 1 and V 2 the partite sets of the graph G, then without loss of generality if In 1967 Rosa proved the following [9]: Theorem 3.6 (Rosa 1967). If a graph G with n edges has an ↵-labeling, then there exists a cyclic decomposition of the complete graph K 2nk+1 into subgraphs isomorphic to G, where k is an arbitrary positive integer.
El-Zanati and Vanden Eynden extended the results of Rosa by defining several new labelings (see, e.g., [4]), which are less restrictive than ↵-labeling but also give decompositions of larger complete graphs.
The only difference between ↵-and near ↵-labelings is that while in ↵-labeling all vertices in Thus, a near ↵-labelings can be seen as a "locally ↵-labeling" as some labels of vertices in V 1 can exceed labels of some vertices in V 2 when they are not neighbors. El-Zanati, Kenig, and Vanden Eynden [5] proved that near ↵-labelings also give decompositions of complete graphs. In fact, we will use an even less restrictive labeling mentioned by El-Zanati and Vanden Eynden in [4] and called + -labeling.
Notice that every near ↵-labeling is also a + -labeling with the additional restriction that the highest label is n. Although the following theorem was never formally proved in [4], we state it nevertheless. The proof is identical to the proof of Theorem 3.8, since the property that all vertex labels are at most n is never used there.
Theorem 3.10. If a bipartite graph G with n edges has a + -labeling, then there exists a Gdecomposition of K 2nk+1 for any positive integer k.
The above labelings enable isomorphic decompositions of complete graphs K m of order m ⌘ 1 (mod 2nk), but similar methods exist for order m ⌘ 0 (mod 2nk) under certain conditions. It is well known that ⇢-labelings can be also used for decompositions of K 2nk .
Theorem 3.11 (see, e.g., [4]). Let G be a graph with n edges and let v be a vertex of degree 1 in G. If G v has a ⇢-labeling, then there exists a G-decomposition of K 2n .
We will prove a similar result for ↵-and + -labelings to obtain our main result.

Necessary conditions and catalog of graphs
Let G be a graph with 8 edges. Since the number of edges in a G-decomposable complete graph K m must be a multiple of the number of edges of G, we have m(m 1)/2 = 8n, which yields m = 16n or m = 16n + 1. Therefore, we only need to investigate graphs K 16n and K 16n+1 for n 1. We also notice that because C 8 is regular of degree 2, it cannot decompose K 16n , since the vertices in the complete graph K 16n are of an odd degree 16n 1. Hence, we have the following. Now we list all connected unicyclic graphs with 8 edges by providing their drawings. There are 34 non-isomorphic unicyclic graphs with an even cycle on 8 vertices (including C 8 ). We first introduce some notation. The type of such graph G will be determined by an l-tuple i 1 , i 2 , . . . , i l , where l is the length of the only cycle C l in G with vertices v 1 , v 2 , . . . , v l and i j is the number of edges in the tree attached to v j . We will always attach the largest tree to v 1 and of course list only one of the two symmetric options.
So for instance, 1, 0, 0, 1, 0, 0 denotes a six-cycle graph with pendant vertices having two vertices attached to v 1 and v 4 , respectively, and 3, 1, 0, 0 denotes four-cycle graphs with an attached tree with 3 edges containing v 1 and a pendant vertex connected to v 2 . Notice that there are four non-isomorphic graphs of this type. We present the graphs in an order that minimizes the space used for the figures and skip the obvious figure of C 8 .

Labelings
In this section we present ↵-and + -labelings for all graphs in our class. For decompositions of complete graphs K 16n+1 , no additional properties would be needed. However, to apply Theorem 3.11, we need a pendant edge of length 8. Once we have it, we can apply the following theorem to justify our main result.
Theorem 5.1. Let G be a bipartite graph on r edges with an ↵or + -labeling such that the longest edge of length r is a pendant edge e. Then there exists a graph H on rk edges that has a ⇢-labeling and can be decomposed into k copies of G and a graph H on rk 1 edges that has a ⇢-labeling and can be decomposed into k 1 copies of G and one copy of G e.
Proof. Let G be a graph as above with bipartition V 0 and V 1 . Without loss of generality, let V 0 comprise {x 0 1 , x 0 2 , . . . , x 0 a } and V 1 comprise {x 1 1 , x 1 2 , . . . , x 1 b }. From the Definitions 3.7 and 3.9 it follows that f (x 0 i ) < f(x 1 j ) whenever x 0 i x 1 j is an edge of G and the lengths of all edges x 0 i x 1 j 2 E(G) form a bijection with {1, 2, . . . , r}. Denote this copy of G by G 1 and assume that the longest edge e of length`(e) = r is x 0 1 x 1 1 where x 1 1 is of degree 1. Now for t = 2, 3, . . . , k take a graph G t , isomorphic to G, with bipartition V 0 and V t = {x t 1 , x t 2 , . . . , x t b }. Add (t 1)r to each vertex label in V 1 of G 1 to label vertices in V t and keep the labels in V 0 the same as before. More precisely, let f (x t j ) = f (x 1 j ) + (t 1)r.

Now the length of an edge
and the edge lengths of G t form the set {(t 1)r + 1, (t 1)r + 2, . . . , tr}. The resulting graph H now can be decomposed into k copies of G and contains edges of lengths 1, 2, . . . , rk. To obtain the graph H , we simply remove the pendant, longest edge x 0 1 x k 1 of length rk. It should be clear from the construction that H can also be decomposed as required. Because all edge lengths were calculated as f (x t j ) f (x 0 i ), it is irrelevant whether the total number of vertices of the complete graph that we are decomposing is 2rk 1 or 2rk + 1.
The following theorem is a direct consequence of Theorems 3.11 and 5.1.
Theorem 5.2. Let G be a graph with an ↵or + -labeling on r edges such that the edge of length r is a pendant edge. Then there exists a G-decomposition of K 2rk for any positive integer k.
Proof. By Theorem 5.1, we can construct a ⇢-labeled graph H consisting of k 1 edge-disjoint copies of G and one copy of G e, where e = uv and v is of degree one. Let H be the graph arising from H by adding back the edge e = uv. Then the graph H with rk edges satisfies the assumptions of Theorem 3.11 and therefore decomposes K 2rk .
Rosa [9] proved that C 4m has an ↵-labeling for any m 1. Hence, the following special case holds by Theorem 3.6. Theorem 5.3 (Rosa 1967). The cycle C 8 has an ↵-labeling and thus decomposes K 16n+1 for every positive integer n.
Next we present ↵-or + -labelings of all connected bipartite unicyclic graphs with 8 edges except C 8 , which we do not need because of the result above.
Notice that the graphs in Figure 18 have a + -labeling, while the remaining ones have an ↵labeling. The number in brackets is just a counter for different non-isomorphic graphs of the same type. For simplicity, in sub-captions we state just the type of the graph and the counter (where needed), omitting the word "type."                We are now ready to prove our main result. Proof. The necessity follows from Proposition 4.1. Figures 13 to 19 show that all graphs in question have ↵-or + -labeling with a pendant edge of length 8. Therefore, they satisfy assumptions of Theorems 3.6 and 5.2 and by these theorems, they decompose every K 16n and K 16n+1 .