Regular handicap graphs of order $n \equiv 0$ (mod 8)

An antimagic labeling of a graph G is a bijection f:E(G)→{1,…,|E(G)|} such that the weights w(x)=∑y∼xf(y) distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1990) is that every connected graph other than K2 admits an antimagic labeling. For a set of distances D, a D-antimagic labeling of a graph G is a bijection f:V(G)→{1,…,|V(G)|} such that the weightω(x)=∑y∈ND(x)f(y) is distinct for each vertex x, where ND(x)={y∈V(G)|d(x,y)∈D} is the D-neigbourhood set of a vertex x. If ND(x)=r, for every vertex x in G, a graph G is said to be (D,r)-regular. In this paper, we conjecture that a graph admits a D-antimagic labeling if and only if it does not contain two vertices having the same D-neighborhood set. We also provide evidence that the conjecture is true. We present computational results that, for D={1}, all graphs of order up to 8 concur with the conjecture. We prove that the set of (D,r)-regular D-antimagic graphs is closed under union. We provide examples of disjoint union of symmetric (D,r)-regular that are D-antimagic and examples of disjoint union of non-symmetric non-(D,r)-regular graphs that are D-antimagic. Furthermore, lastly, we show that it is possible to obtain a D-antimagic graph from a previously known distance antimagic graph.

In this paper it is shown that if a graph G possesses a spanning subgraph H with a strong vertex magic total labeling (VMTL) and $G-E(H)$ is even‐regular, then G also has a strong VMTL. Among other things, this is used to conclude that all Hamiltonian regular graphs of odd order possess strong VMTLs. A relationship is then demonstrated between regular graphs of even degree and sparse magic squares. We next consider cubic graphs of order $2n$ consisting of two 2‐factors of order n, connected by a 1‐factor (quasi‐prisms). Based on McQuillan’s construction of VMTLs of such 3‐regular graphs, VMTLs are derived for similar regular graphs of any odd degree. Finally, a construction is given for VMTLs of quartic graphs of order $4n+2$ consisting of two cycles of odd order n connected by a 2‐factor (simple quasi‐anti‐prisms), and based on this construction VMTLs are derived for similar regular graphs of any even degree.

Vertex-magic labelings of regular graphs II

Hamiltonian Cycles in Regular Graphs

Let be a graph of order n. Let be a bijection. For any vertex , the neighbor sum is called the weight of the vertex and is denoted by where N(v) is the open neighborhood of If for any two distinct vertices and then f is called a distance antimagic labelling. If the graph G admits such a labelling, then G is said to be a distance antimagic graph. This study gives a distance antimagic labelling for tensor product of two complete graph and sufficient condition so that the tensor product of a regular graph and complete graph is a distance antimagic graph.

Pseudo-one-factorizations of regular graphs of odd order — II. products of graphs

A complete classification is given of 2-distance-transitive circulants, which shows that a 2-distance-transitive circulant is a cycle, a Paley graph of prime order, a regular complete multipartite graph, or a regular complete bipartite graph of order twice an odd integer minus a 1-factor.

A 1-factorisation of a graph G is a decomposition of G into edge-disjoint 1-factors (1-regular spanning subgraphs). A perfect 1-factorisation is a 1-factorisation in which the union of every pair of distinct 1-factors is a Hamilton cycle; a 1-factorisation is uniform if the graphs resulting from the union of every pair of distinct 1-factors are isomorphic. We initiate the study of perfect 1-factorisations and uniform 1-factorisations of circulant graphs, a well-studied family of Cayley graphs, by determining whether or not such 1-factorisations exist for even order circulant graphs of small degree. For a bipartite circulant graph, we observe that having order congruent to 2 (mod 4) is a necessary condition for the existence of a perfect 1-factorisation. We prove that for bipartite 3-regular circulant graphs this necessary condition is also sufficient. Furthermore, the bipartite 3-regular circulant graphs of order 2 (mod 4) are the only 3-regular circulant graphs of order greater than 6 which admit a perfect 1-factorisation. The 4-regular case is more complex; we provide constructions of perfect 1-factorisations for many families of bipartite 4-regular circulant graphs of order 2 (mod 4); however, we also prove that there is an infinite family of such graphs no member of which admits a perfect 1-factorisation. We conjecture that a connected non-bipartite 4-regular circulant graph of order at least 8 does not admit a perfect 1-factorisation; we prove this conjecture for three distinct infinite families of non-bipartite 4-regular circulant graphs and show by computer search that the conjecture holds for small orders (at most 30). Several of our results are achieved by studying perfect 1-factorisations of a particular family of 4-regular graphs, denoted Dh, k for integers h, knwhere h g 2 and k g 3. These graphs are shown to be isomorphic to Cayley graphs when k is even and yet they are not vertex-transitive when k is odd (and hng 3). For small values of h, we make use of the structure of Dh,k to determine when it admits a perfect 1-factorisation and then use specific isomorphism results to obtain corresponding results for 4-regular circulant graphs. In this thesis the connected 3-regular circulant graphs that admit a perfect 1-factorisation are characterised and although the connected 4-regular circulant graphs that admit a perfect 1-factorisation have not yet been fully characterised, we provide many interesting theoretical results that establish the framework for such a characterisation.n

Let G be a graph of order n. A bijection $$f:VG \longrightarrow \{1, 2, \ldots , n\}$$ is said to be distance antimagic if for every vertex v the vertex weight defined by $$w_fv =\sum _{x\in Nv}fx$$ is distinct. The graph which admits such a labeling is called a distance antimagic graph. For a positive integer k, define $$f_{k}:VG \longrightarrow \{1+k,2+k, \ldots , n+k\}$$ by $$f_{k}x=fx+k $$. If $$w_{f_{k}}u\ne w_{f_{k}}v$$ for every pair of vertices $$u,v \in V$$, for any $$k\ge 0$$ then f is said to be an arbitrarily distance antimagic labeling and the graph which admits such a labeling is said to be an arbitrarily distance antimagic graph. In this paper, we provide arbitrarily distance antimagic labelings for $$rP_n$$, generalised Petersen graph $$Pn,k, n \ge 5$$, Harary graph $$H_{4,n}$$ for $$n\ne 6$$ and also prove that join of these graphs is distance antimagic.

Constructing an Infinite Family of Cubic 1-Regular Graphs

Upper bounds for independent domination in regular graphs

On the non-existence of antipodal cages of even girth

Long cycles through specified vertices in a graph

Interference patterns in bijective colorings of 2-regular graphs

The Moore bound $M(k,g)$ is a lower bound on the order of $k$-regular graphs of girth $g$ (denoted $(k,g)$-graphs). The excess $e$ of a $(k,g)$-graph of order $n$ is the difference $ n-M(k,g) $. In this paper we consider the existence of $(k,g)$-bipartite graphs of excess $4$ by studying spectral properties of their adjacency matrices. For a given graph $G$ and for the integers $i$ with $0\leq i\leq diam(G)$, the $i$-distance matrix $A_i$ of $G$ is an $n\times n$ matrix such that the entry in position $(u,v)$ is $1$ if the distance between the vertices $u$ and $v$ is $i$, and zero otherwise. We prove that the $(k,g)$-bipartite graphs of excess $4$ satisfy the equation $kJ=(A+kI)(H_{d-1}(A)+E)$, where $A=A_{1}$ denotes the adjacency matrix of the graph in question, $J$ the $n \times n$ all-ones matrix, $E=A_{d+1}$ the adjacency matrix of a union of vertex-disjoint cycles, and $H_{d-1}(x)$ is the Dickson polynomial of the second kind with parameter $k-1$ and degree $d-1$. We observe that the eigenvalues other than $\pm k$ of these graphs are roots of the polynomials $H_{d-1}(x)+\lambda$, where $\lambda$ is an eigenvalue of $E$. Based on the irreducibility of $H_{d-1}(x)\pm2$, we give necessary conditions for the existence of these graphs. If $E$ is the adjacency matrix of a cycle of order $n$, we call the corresponding graphs graphs with cyclic excess; if $E$ is the adjacency matrix of a disjoint union of two cycles, we call the corresponding graphs graphs with bicyclic excess. In this paper we prove the non-existence of $(k,g)$-graphs with cyclic excess $4$ if $k\geq6$ and $k \equiv1 \!\! \pmod {3}$, $g=8, 12, 16$ or $k \equiv2 \!\! \pmod {3}$, $g=8;$ and the non-existence of $(k,g)$-graphs with bicyclic excess $4$ if $k\geq7$ is an odd number and $g=2d$ such that $d\geq4$ is even.