Decomposition of complete graphs into connected unicyclic graphs with eight edges and pentagon

A G-decomposition of the complete graph Kn is a family of pairwise edge disjoint subgraphs of Kn, all isomorphic to G, such that every edge of Kn belongs to exactly one copy of G. Using standard decomposition techniques based on ρ-labelings, introduced by Rosa in 1967, and their modifications we show that each of the ten non-isomorphic connected unicyclic graphs with eight edges containing the pentagon decomposes the complete graph Kn whenever the necessary conditions are satisfied.


Introduction
A decomposition of the complete graph K n is a family of pairwise edge disjoint subgraphs D = {G 0 , G 1 , . . ., G s } such that every edge of K n belongs to exactly one member of D. If each subgraph G r is isomorphic to a given graph G we speak about a G-decomposition of K n , or a G-design.The decomposition is cyclic if there exists an ordering (x 1 , x 2 , . . ., x n ) of the vertices of K n and isomorphisms φ r : G 0 → G r , r = 0, 1, 2, . . ., s, such that φ r (x i ) = x i+r for each i = 1, 2, . . ., n.The subscripts are taken modulo n.Similarly, the decomposition is one-rotational if there exists an ordering (x 1 , x 2 , . . ., x n ) of the vertices of K n and isomorphisms φ r : G 0 → G r , www.ijc.or.idDecomposition of K n into connected unicyclic graphs with 8 edges ... | D. Froncek and O. Kingston r = 0, 1, 2, . . ., s, such that φ r (x i ) = x i+r for each i = 1, 2, . . ., n − 1 and φ r (x n ) = x n .The subscripts here are taken modulo n − 1.
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 ten connected unicyclic graphs with eight edges containing the pentagon and prove that each of them decomposes the complete graph K n whenever the necessary conditions are satisfied.
We will use standard decomposition methods based on ρ-labelings, introduced by Rosa [8] as well as some other slightly modified well-known tools.

Related results
It seems that there has been no attempt to completely classify graphs with 8 vertices and 8 edges decomposing complete graphs.Between the time the results contained in this paper were obtained and prepared for publication, a partial result on four particular connected graphs with pentagon was published in [10].The methods used in the paper were mostly design theory based and are significantly different from methods that we use in this paper.Therefore, for the sake of completeness, we include our results for those graphs as well.
At the same time, there are several papers on decompositions of K n into graphs with 8 edges and 5, 6 or 7 vertices.This section will summarize what is known about classification of smaller graphs, that is, graphs where |(E(G)| ≤ 8 or |(V (G)| ≤ 8 that form decompositions of complete graphs.An exhaustive search was conducted, and the following results were found, however, due to the limited availability of some journals there could be other results not summarized here.
Graphs with five vertices and eight edges were examined by Colbourn, Ge, and Ling in 2008, due to their applicability with respect to the problem of grooming traffic in optical networks [3].There are only two non-isomorphic graphs with five vertices and eight edges, shown in Figure 1.Colbourn, Ge and Ling proved the following results for the graphs G 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 2.3 (Kang, Yuan, Liu 2005).Let G be a connected graph with six vertices and eight edges.Then G forms a decomposition of K n if and only if n ≡ 0, 1 (mod 16) and n ≥ 16 with two possible exceptions for n = 32.
We were unable to find any results on graphs with 8 edges and 7 vertices.For graphs with 8 edges and 8 vertices, Kang and Zhang determined the spectrum completely for the four graphs shown in Figure 2. Bipartite connected unicyclic graphs with 8 edges and 8 vertices other than C 8 were studied by Fahnenstiel and Froncek in [5]; C 8 decompositions exist if and only if n ≡ 1 (mod 16) as proved by Rosa [8].
Theorem 2.5 (Fahnenstiel, Froncek 2017).Let G be a connected bipartite unicyclic graph with eight vertices and eight edges other than C 8 .Then G forms a decomposition of K n if and only if n ≡ 0, 1 (mod 16) and n ≥ 16.

Tools and methods
The major tool we use is the ρ-labeling, first defined by Rosa [8], although he originally used the term valuation.
A more restrictive version is the σ-labeling, also introduced by Rosa in [8].
Rosa [8] proved that if a graph G with n edges has one of the above labelings, then a decomposition of a complete graph K 2n+1 exists.Theorem 3.1 (Rosa 1967).A cyclic decomposition of the complete graph K 2n+1 into subgraphs isomorphic to a given graph G with n edges exists if and only if there exists a ρ-labeling of the graph G.
The above labelings enable isomorphic decompositions of complete graphs of odd order, but similar methods exist for complete graphs of even order under certain circumstances.It is well known that certain ρ-labeled graphs can form isomorphic decompositions of K 2nk (see, e.g., [4]).Although the proof of the theorem is well-known, we repeat it here for the reader's convenience.Theorem 3.2.Let G be a graph with n edges and let v be a vertex of degree 1 in G.If G − v has a ρ-labeling, then G forms a one-rotational isomorphic decomposition of K 2n .
Proof.Let uv be the only edge incident with vertex v in G.We denote the vertices of K 2n as x 1 , x 2 , . . ., x 2n .
Because G − v allows a ρ-labeling, by Theorem 3.2 there exists a cyclic decomposition of We can assume without loss of generality that in the i-th copy of G − v, G i , the vertex u is mapped onto x i , and identify v with x 2n .Then for each i = 1, 2, . . ., 2n − 1, the edge x i x 2n of K 2n appears exactly in one copy of G − v, namely in G i .Obviously, adding the edge x i x 2n = uv to the appropriate G i , we obtain a G-decomposition of K 2n .The length of uv will be denoted by ∞.
For bipartite graphs, Rosa also found another labeling (called α-labeling) that allows decompositions of larger complete graphs, namely K 2nk+1 for any positive integer k.This labeling cannot be used for graphs with odd cycles.However, one can observe that all unicyclic graphs are almost bipartite, that is, we can obtain a bipartite graph by removing a single edge.Blinco, El-Zanati, and Vanden Eynden [1] provide a similar tool, γ-labeling, that allows decomposition of K 2nk+1 into almost bipartite graphs.Even more powerful labeling for tripartite graphs, called tripartite ρ-labeling, was later found by Bunge, Chantasartrassmee, El-Zanati, and Vanden Eynden [2].
It would be probably possible to find a γor tripartite ρ-labeling for most or all of the graphs we are focusing on, but even then we would have to use another method for decompositions of K 2nk .Therefore, we decided to use a different approach that will cover decompositions of both K 2nk+1 and K 2nk concurrently.
We will find a σ-labeling for a graph H satisfying assumptions of Theorem 3.2 that can be decomposed into k copies of G, where G is a connected unicyclic graph with 8 edges containing a pentagon.Because we use a σ-labeling rather than ρ-labeling, the lengths of edges are always computed as (uw) = |f (u) − f (w)| regardless of whether we are using the lengths with respect to K 2n+1 or K 2n−1 and we can replace the longest edge uv by the edge of length ∞.
We formalize the method in the following observation.Proposition 3.3.Let H be a graph decomposable into k copies of a graph G and there exists an H-decomposition of K m .Then there also exists a G-decomposition of K m .www.ijc.or.idWhile for n = 16k + 1 we could use even labels 16k − 1 and 16k (which is the largest label in K 16k+1 ), for n = 16k the vertex x 16k is labeled ∞ and for x 1 , x 2 , . . ., x 16k−1 we use labels 0, 1, . . ., 16k − 2. Construction 4.1 (k ≥ 2).We first label the pentagons with one attached edge as shown in Fig- ure 4. Vertex labels are in blue, edge lengths in red.Notice that in the k-th copy G k the largest label is not exceeding 16k − 2, which is the maximum allowed in K 16k .The range of edge lengths used in these subgraphs is from 2k to 8k − 1.The remaining lengths 1, 2, . . ., 2k − 1 and 8k will be used for the additional edges.Now we list the remaining edges for the particular graphs.Notice that while the first k − 1 graphs G 1 , G 2 , . . ., G k−1 always follow the same pattern, the last copy G k is different, since we need the longest edge to have one end-vertex of degree one.When k = 1, we just disregard the first two rows and read only the rows for G k .
The asterisk denotes that the vertex label or edge length needs to be replaced by ∞ for decompositions of K 16 .

Conclusion
In this paper, we have taken further steps to the complete characterization of graphs with eight edges decomposing complete graphs.We have completely solved the problem of decomposition of the complete graph K n into connected unicyclic graphs with eight edges and the unique cycle having length five.
In an earlier paper, Fahnenstiel and Froncek [5] solved analogical problem for connected bipartite unicyclic graphs with eight edges.The remaining cases of unicyclic graphs with eight edges, both connected and disconnected, have been recently solved by Freyberg and Froncek [6].Thus, the class of unicyclic graphs with eight edges has been completely classified.

Figure 4 :
Figure 4: Pentagon with pendant edge in k-th copy Decomposition of K n into connected unicyclic graphs with 8 edges ... | D. Froncek and O. Kingston

5 * 8k *
Thus we have shown that for every connected unicyclic graph G(b 1 , b 2 , . . ., b 5 ; t) on eight edges with a pentagon and every k ≥ 1 there exists a corresponding graph H(b 1 , b 2 , . . ., b 5 ; t) decomposable into k copies of G(b 1 , b 2 , . . ., b 5 ; t) with a σ-labeling and longest edge having an end-vertex of degree one.Our main result now follows easily.Theorem 4.1.Let G be a connected unicyclic graph with eight vertices and eight edges where the unique cycle is a pentagon.Then G forms a decomposition of K n if and only if n ≡ 0, 1 (mod 16) and n ≥ 16.Proof.By Construction 4.1, for every connected unicyclic graph on eight edges with a pentagon G(b 1 , b 2 , . . ., b 5 ; t) and every k ≥ 1 there exists a corresponding graph H(b 1 , b 2 , . . ., b 5 ; t) decomposable into k copies of the graph G(b 1 , b 2 , . . ., b 5 ; t).Moreover, each such H(b 1 , b 2 , . . ., b 5 ; t) has a σ-labeling and the longest edge has an end-vertex of degree one.Thus, by Theorems 3.1 and 3.2, each H(b 1 , b 2 , . . ., b 5 ; t) decomposes K n for every n ≡ 0, 1 (mod 16) and n ≥ 16.Because each H(b 1 , b 2 , . . ., b 5 ; t) is decomposable into k copies of the corresponding graph G(b 1 , b 2 , . . ., b 5 ; t), the result now follows by Proposition 3.3.