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AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 75(3) (2019), Pages 343{356 Graceful pairings Jocelyn R. Bell Department of Mathematics and Computer Science Hobart and William Smith Colleges Geneva, NY 14456 U.S.A. [email protected] Desmond F. Cummins Department of Computer Science Wells College Aurora, NY 13026 U.S.A. [email protected] Abstract A graceful labeling of a graph G with n edges is an injection from the set of vertices of G to f0; 1; : : : ; ng such that if each edge of G is labeled by the absolute value of the difference of the labels of its incident vertices, then every edge has a distinct label in f1; : : : ; ng. The famously unsettled graceful labeling conjecture proposes that every tree has a graceful labeling. A graceful labeling θ of a graph G is said to be gracious if for each vertex v of G either all adjacent vertices have larger labels than θ(v) or all adjacent vertices have smaller labels than θ(v). We introduce novel machinery for combining graceful bipartite graphs to produce new graceful graphs. If the constituent graphs have a gracious labeling then our methods produce a gracious labeling. Infinite families of gracious trees are produced and new classes of graceful trees are introduced. Along the way we offer a partial solution to a question posed in 1979. 1 Introduction Graceful labelings of graphs were introduced by A. Rosa in 1967 [19]. These labelings have practical as well as theoretical applications. For example, graceful labelings played a central role in the solution of the two-table Oberwolfach problems [21]. A graceful graph with n edges cyclically decomposes the complete graph on 2n + 1 ISSN: 2202-3518 c The author(s). Released under the CC BY 4.0 International License J.R. BELL AND D.F. CUMMINS / AUSTRALAS. J. COMBIN. 75 (3) (2019), 343{356 344 Figure 1: A gracious tree which cannot be α-labeled vertices [19]. There are graphs which are not graceful but it is unknown whether or not every tree is graceful. The graceful tree conjecture proposes that every tree is graceful. While many papers have been written on this conjecture [9] it remains unresolved. Some general methods for generating graceful labelings have been discovered [9]; for some previous constructions, see [5] and [13]. The \product" method presented in [13] (in fact, it appears earlier in [20]) has been subsequently modified and extended many times (see [3, 14, 15]). In this paper we present new methods for combining certain families of gracefully labeled graphs to produce new gracefully labeled graphs. These theorems are of particular interest since our construction is fundamentally different from previous constructions. A graph G is a set of vertices V (G) together with a set of edges E(G) ⊆ V (G) × V (G) whose elements are denoted uv. All graphs are assumed to be simple and connected. If G is a graph and v is a vertex of G, the neighborhood of v in G, denoted by N(v); is the set of all vertices adjacent to v.A tree is a connected graph with no cycles. We refer the reader to West [22] for definitions omitted. A labeling of a graph G is an injection θ : V (G) ! f0; 1; 2;:::g.A labeled graph is a pair (G; θ) where G is a graph and θ is a labeling of G. When θ is clear from context, we may refer to the labeled graph (G; θ) as G and the vertex v of G which is labeled k as the k-vertex of G. Every labeling θ of a graph G induces a labeling on E(G) in the following manner: if uv is an edge of G, assign the induced edge label θ(uv) := jθ (u) − θ (v)j to the edge uv. A labeling θ of a graph G with n edges is graceful if the range of θ is contained in the set f0; 1; : : : ; ng and the set of induced edge labels is precisely f1; 2; : : : ; ng. If a graph G admits a graceful labeling then G is said to be graceful. All trees with at most 35 vertices are claimed to be graceful in [8]. For a comprehensive survey of known results on the topic of graceful labeling see [9]. A labeling θ of a graph G with n edges is said to be an α-labeling (originally termed an α-valuation by Rosa [19]) if there exists a natural number s such that for all edges uv of G, either θ(u) ≤ s < θ(v) or θ(v) ≤ s < θ(u). The number s is the boundary value of the α-labeling. In this paper Pn will denote the path with n edges and n + 1 vertices. All paths and caterpillar graphs (paths with any number of additional pendant vertices) can be α-labeled [19]. However, there are graceful trees for which an α-labeling does not exist. The smallest example of such a tree is the \spider" with 3 legs, each of which has two edges: see Figure 1. A labeling θ of a graph G is said to be ordered if for each v 2 V (G), either for all J.R. BELL AND D.F. CUMMINS / AUSTRALAS. J. COMBIN. 75 (3) (2019), 343{356 345 u 2 N(v), θ(u) < θ(v) or for all u 2 N(v), θ(u) > θ(v). This definition appears to be due to Cahit [6]; however the concept is also introduced in [10] in which it is called a gracious labeling and a graph admitting a gracious labeling is termed gracious. It is also rediscovered in [7], therein referred to as a near-α labeling. In this paper we will use the term gracious. Every α-labeled tree is also a gracious tree, but not conversely; see Figure 1. It is well-known that the tree in Figure 1 has no α-labeling. Every tree with at most 20 vertices has been verified to have a gracious labeling [10]. The ordered (or near-α or gracious) labeling conjecture proposes that every tree has an ordered labeling [6, 7, 10]. Every graph with k edges with a gracious labeling cyclically decomposes the complete graph on 2kn + 1 vertices [7]. A graph G is bipartite if there exists a partition (A; B) of V (G) such that for each edge e of G, e = uv where u 2 A and v 2 B. If G admits an α-labeling or a gracious labeling then G is bipartite. Not all bipartite graphs are graceful; for example the cycle graph C6 is bipartite yet fails to be graceful [19]. If (G; θ) is a gracefully labeled graph with n edges, the complementary labeling θ¯ of G is defined for each vertex v of G by θ¯(v) = n − θ(v). If θ is a graceful labeling, an α-labeling, or a gracious labeling, then so is θ:¯ If f is a function, ranf will denote the range of f. If (G; θ) is a labeled graph, a relabeling function is a function f : ranθ ! f0; 1; 2;:::g. We may abuse notation and denote (G; f ◦ θ) by (G; f). Suppose (G; θ) is a gracefully labeled bipartite graph with bipartition (A; B). We will say an edge e of G is positively oriented (with respect to the bipartition (A; B)) provided e = uv where u 2 A, v 2 B and θ(u) > θ(v). The labeling θ is gracious if and only if the set of positively oriented edges of G is either empty or is equal to E(G). Definition 1.1. Suppose C = f(G1; θ1) ;:::; (Gm; θm)g is an indexed collection of disjoint bipartite gracefully labeled graphs each with n edges. C is a compatible collection if there exists a partition (A; B) of f0; 1; : : : ; ng and for each i with 1 ≤ 0 0 0 i ≤ m there exists a bipartition (Ai;Bi) of V (Gi) so that if Ai = fθi(v): v 2 Aig, 0 Bi = fθi(v): v 2 Big and Pi = fθi(uv): uv is a positively oriented edge of Gig ; then for each i, Ai ⊆ A, Bi ⊆ B and Pi = P1. If C is a compatible collection in which all graphs are graciously labeled, then C is a gracious compatible collection. See Figure 2 for an example of a gracious compatible collection for which one may choose A = f3; 4; 6; 7g, B = f0; 1; 2; 5g, and 0 Bi to be the set of vertices of Gi colored white. Definition 1.2. Suppose C = f(G1; θ1) ;:::; (Gm; θm)g is a compatible collection of graphs. For each i with 1 ≤ i ≤ m, let wi be the 0-vertex of Gi and let Zi = fθi(vwi): vwi 2 E(Gi)g. C is a 0-compatible collection if for each i, Zi = Z1. See Figure 3 for an example of a 0-compatible collection of graphs each with 7 edges for which Z1 = f5; 7g. J.R. BELL AND D.F. CUMMINS / AUSTRALAS. J. COMBIN. 75 (3) (2019), 343{356 346 Figure 2: A gracious compatible collection Figure 3: A 0-compatible collection We introduce graceful pairings in Section 2. In Section 3 we prove that if m = 2k + 1, C = f(G1; θ1) ;:::; (Gm; θm)g is a (gracious) 0-compatible collection, and G is a (graciously) gracefully labeled graph with k edges, then the graph formed by identifying the graphs G; G1;:::;Gm at their 0-vertices is (gracious) graceful.
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