On d-graceful labelings

In this paper we introduce a generalization of the well known concept of a graceful labeling. Given a graph G with e=dm edges, we call d-graceful labeling of G an injective function from V(G) to the set {0,1,2,..., d(m+1)-1} such that {|f(x)-f(y)| | [x,y]\in E(G)} ={1,2,3,...,d(m+1)-1}-{m+1,2(m+1),...,(d-1)(m+1)}. In the case of d=1 and of d=e we find the classical notion of a graceful labeling and of an odd graceful labeling, respectively. Also, we call d-graceful \alpha-labeling of a bipartite graph G a d-graceful labeling of G with the property that its maximum value on one of the two bipartite sets does not reach its minimum value on the other one. We show that these new concepts allow to obtain certain cyclic graph decompositions. We investigate the existence of d-graceful \alpha-labelings for several classes of bipartite graphs, completely solving the problem for paths and stars and giving partial results about cycles of even length and ladders.Comment: In press on Ars Combi

Decomposition of Certain Complete Graphs and Complete Multipartite Graphs into Almost-bipartite Graphs and Bipartite Graphs

In his classical paper [14], Rosa introduced a hierarchical series of labelings called ρ, σ, β and α labeling as a tool to settle Ringel’s Conjecture which states that if T is any tree with m edges then the complete graph K2m+1 can be decomposed into 2m + 1 copies of T . Inspired by the result of Rosa [14] many researchers significantly contributed to the theory of graph decomposition using graph labeling. In this direction, in 2004, Blinco et al. [6] introduced γ-labeling as a stronger version of ρ-labeling. A function g defined on the vertex set of a graph G with n edges is called a γ-labeling if (i) g is a ρ-labeling of G, (ii) G is a tripartite graph with vertex tripartition (A, B, C) with C = {c} and ¯b ∈ B such that {¯b, c} is the unique edge joining an element of B to c, (iii) g(a) \u3c g(v) for every edge {a, v} ∈ E(G) where a ∈ A, (iv) g(c) - g(¯b) = n. Further, Blinco et al. [6] proved a significant result that the complete graph K2cn+1 can be cyclically decomposed into c(2cn + 1) copies of any γ-labeled graph with n edges, where c is any positive integer. Recently, in 2013, Anita Pasotti [4] introduced a generalisation of graceful labeling called d-divisible graceful labeling as a tool to obtain cyclic G-decompositions in complete multipartite graphs. Let G be a graph of size e = d . m. A d-divisible graceful labeling of the graph G is an injective function g : V (G) → {0, 1, 2, . . . , d(m + 1) - 1} such that {|g(u) - g(v)|/{u, v} ∈ E(G)} = {1, 2, . . . , d(m + 1) - 1}\{m + 1, 2(m + 1), . . . , (d - 1)(m + 1)}. A d-divisible graceful labeling of a bipartite graph G is called as a d-divisible α-labeling of G if the maximum value of one of the two bipartite sets is less than the minimum value of the other one. Further, Anita Pasotti [4] proved a significant result that the complete multipartite graph K (e/d +1)×2dc can be cyclically decomposed into copies of d-divisible α-labeled graph G, where e is the size of the graph G and c is any positive integer (K (e/d +1)×2dc contains e/d + 1 parts each of size 2dc). Motivated by the results of Blinco et al. [6] and Anita Pasotti [4], in this paper we prove the following results. i) For t ≥ 2, disjoint union of t copies of the complete bipartite graph Km,n, where m≥ 3, n ≥ 4 plus an edge admits γ-labeling. ii) For t ≥ 2, t-levels shadow graph of the path Pdn+1 admits d-divisible α-labeling for any admissible d and n ≥ 1. Further, we discuss related open problems

Decompositions of Mixed Graphs with Partial Orientations of the P\u3csub\u3e4\u3c/sub\u3e.

A decomposition D of a graph H by a graph G is a partition of the edge set of H such that the subgraph induced by the edges in each part of the partition is isomorphic to G. A mixed graph on V vertices is an ordered pair (V,C), where V is a set of vertices, |V| = v, and C is a set of ordered and unordered pairs, denoted (x, y) and [x, y] respectively, of elements of V [8]. An ordered pair (x, y) ∈ C is called an arc of (V,C) and an unordered pair [x, y] ∈ C is called an edge of graph (V,C). A path on n vertices is denoted as Pn. A partial orientation on G is obtained by replacing each edge [x, y] ∈ E(G) with either (x, y), (y, x), or [x, y] in such a way that there are twice as many arcs as edges. The complete mixed graph on v vertices, denoted Mv, is the mixed graph (V,C) where for every pair of distinct vertices v1, v2 ∈ V , we have {(v1, v2), (v2, v1), [v1, v2]} ⊂ C. The goal of this thesis is to establish necessary and sufficient conditions for decomposition of Mv by all possible partial orientations of P4

A survey on constructive methods for the Oberwolfach problem and its variants

The generalized Oberwolfach problem asks for a decomposition of a graph $G$ into specified 2-regular spanning subgraphs $F_1,\ldots, F_k$, called factors. The classic Oberwolfach problem corresponds to the case when all of the factors are pairwise isomorphic, and $G$ is the complete graph of odd order or the complete graph of even order with the edges of a $1$-factor removed. When there are two possible factor types, it is called the Hamilton-Waterloo problem. In this paper we present a survey of constructive methods which have allowed recent progress in this area. Specifically, we consider blow-up type constructions, particularly as applied to the case when each factor consists of cycles of the same length. We consider the case when the factors are all bipartite (and hence consist of even cycles) and a method for using circulant graphs to find solutions. We also consider constructions which yield solutions with well-behaved automorphisms.Comment: To be published in the Fields Institute Communications book series. 23 pages, 2 figure

A complete solution to the infinite Oberwolfach problem

Let $F$ be a $2$-regular graph of order $v$. The Oberwolfach problem, $OP(F)$, asks for a $2$-factorization of the complete graph on $v$ vertices in which each $2$-factor is isomorphic to $F$. In this paper, we give a complete solution to the Oberwolfach problem over infinite complete graphs, proving the existence of solutions that are regular under the action of a given involution free group $G$. We will also consider the same problem in the more general contest of graphs $F$ that are spanning subgraphs of an infinite complete graph $\mathbb{K}$ and we provide a solution when $F$ is locally finite. Moreover, we characterize the infinite subgraphs $L$ of $F$ such that there exists a solution to $OP(F)$ containing a solution to $OP(L)$

Distance magic-type and distance antimagic-type labelings of graphs

Generally speaking, a distance magic-type labeling of a graph G of order n is a bijection f from the vertex set of the graph to the first n natural numbers or to the elements of a group of order n, with the property that the weight of each vertex is the same. The weight of a vertex x is defined as the sum (or appropriate group operation) of all the labels of vertices adjacent to x. If instead we require that all weights differ, then we refer to the labeling as a distance antimagic-type labeling. This idea can be generalized for directed graphs; the weight will take into consideration the direction of the arcs. In this manuscript, we provide new results for d-handicap labeling, a distance antimagic-type labeling, and introduce a new distance magic-type labeling called orientable Gamma-distance magic labeling. A d-handicap distance antimagic labeling (or just d-handicap labeling for short) of a graph G=(V,E) of order n is a bijection f from V to {1,2,...,n} with induced weight function w(x_{i})=\underset{x_{j}\in N(x_{i})}{\sum}f(x_{j}) \] such that f(x_{i})=i and the sequence of weights w(x_{1}),w(x_{2}),...,w(x_{n}) forms an arithmetic sequence with constant difference d at least 1. If a graph G admits a d-handicap labeling, we say G is a d-handicap graph. A d-handicap incomplete tournament, H(n,k,d) is an incomplete tournament of n teams ranked with the first n natural numbers such that each team plays exactly k games and the strength of schedule of the ith ranked team is d more than the i+1st ranked team. That is, strength of schedule increases arithmetically with strength of team. Constructing an H(n,k,d) is equivalent to finding a d-handicap labeling of a k-regular graph of order n. In Chapter 2 we provide general constructions for every d at least 1 for large classes of both n and k, providing breadth and depth to the catalog of known H(n,k,d)\u27s. In Chapters 3 - 6, we introduce a new type of labeling called orientable Gamma-distance magic labeling. Let Gamma be an abelian group of order n. If for a graph G=(V,E) of order n there exists an orientation of G and a companion bijection f from V to Gamma with the property that there is an element mu in Gamma (called the magic constant) such that \[ w(x)=\sum_{y\in N_{G}^{+}(x)}\overrightarrow{f}(y)-\sum_{y\in N_{G}^{-}(x)}\overrightarrow{f}(y)=\mu for every x in V where w(x) is the weight of vertex x, we say that G is orientable Gamma-distance magic}. In addition to introducing the concept, we provide numerous results on orientable Z_n distance magic graphs, where Z_n is the cyclic group of order n. In Chapter 7, we summarize the results of this dissertation and provide suggestions for future work

Combinatorics

Combinatorics is a fundamental mathematical discipline that focuses on the study of discrete objects and their properties. The present workshop featured research in such diverse areas as Extremal, Probabilistic and Algebraic Combinatorics, Graph Theory, Discrete Geometry, Combinatorial Optimization, Theory of Computation and Statistical Mechanics. It provided current accounts of exciting developments and challenges in these fields and a stimulating venue for a variety of fruitful interactions. This is a report on the meeting, containing extended abstracts of the presentations and a summary of the problem session