On 2-distance-transitive circulants

A vertex $k$-coloring of a graph $G$ is a \emph {multiset $k$-coloring} if $M(u)\neq M(v)$ for every edge $uv\in E(G)$, where $M(u)$ and $M(v)$ denote the multisets of colors of the neighbors of $u$ and $v$, respectively. The minimum $k$ for which $G$ has a multiset $k$-coloring is the \emph {multiset chromatic number} $\chi _{m}(G)$ of $G$. For an integer $\ell \geq 0$, the $\ell $-\emph {corona} of a graph $G$, ${\rm cor}^{\ell }(G)$, is the graph obtained from $G$ by adding, for each vertex $v$ in $G$, $\ell $ new neighbors which are end-vertices. In this paper, the multiset chromatic numbers are determined for \mbox {$\ell $-\emph {coronas}} of all complete graphs, the regular complete multipartite graphs and the Cartesian product $K_{r}\square K_2$ of $K_r$ and $K_2$. In addition, we show that the minimum $\ell $ such that $\chi _{m}({\rm cor}^{\ell }(G))=2$ never exceeds $\chi (G)-2$, where $G$ is a regular graph and $\chi (G)$ is the chromatic number of $G$.

For the set of graphs with a given degree sequence, consisting of any number of $2's$ and $1's$, and its subset of bipartite graphs, we characterize the optimal graphs who maximize and minimize the number of $m$-matchings. We find the expected value of the number of $m$-matchings of $r$-regular bipartite graphs on $2n$ vertices with respect to the two standard measures. We state and discuss the conjectured upper and lower bounds for $m$-matchings in $r$-regular bipartite graphs on $2n$ vertices, and their asymptotic versions for infinite $r$-regular bipartite graphs. We prove these conjectures for $2$-regular bipartite graphs and for $m$-matchings with $m\le 4$.

Eigenvalues and forbidden subgraphs I

In this paper, we consider the minimum maximal matching problem in some classes of graphs such as regular graphs. We show that the minimum maximal matching problem is NP-hard even in regular bipartite graphs, and a polynomial time exact algorithm is given for almost complete regular bipartite graphs. From the approximation point of view, it is well known that any maximal matching guarantees the approximation ratio of 2 but surprisingly very few improvements have been obtained. In this paper we give improved approximation ratios for several classes of graphs. For example any algorithm is shown to guarantee an approximation ratio of (2-o(1)) in graphs with high average degree. We also propose an algorithm guaranteeing for any graph of maximum degree Δ an approximation ratio of (2−1/Δ), which slightly improves the best known results. In addition, we analyse a natural linear-time greedy algorithm guaranteeing a ratio of (2−23/18k) in k-regular graphs admitting a perfect matching.

R. Haggkvist proved that every 3-regular bipartite graph of order 2n with no component isomorphic to the Heawood graph decomposes the complete bipartite graph K 6n,6n . In [2] the first two authors established a necessary and sufficient condition for the existence of a factorization of the complete bipartite graph K n,n into certain families of 3-regular graphs of order 2n. In this paper we tackle the problem of decompositions of K n,n into 3-regular graphs some more. We will show that certain families of 3-regular graphs of order 2n decompose the complete bipartite graph $K_{\frac{3n}{2},\frac{3n}{2}}$.

Regular graphs with girth at least 5 and small second largest eigenvalue

Periodicity of Grover walks on bipartite regular graphs with at most five distinct eigenvalues

Sharp upper bounds for the Laplacian graph eigenvalues

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

Finite common coverings of pairs of regular graphs

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.

We consider the Gathering problem where a swarm of weak robots disposed on the vertices of an anonymous graph are required to meet at one vertex from where they do not move anymore. In our recent work [Cicerone et al., SIROCCO’19], we have shown how synchronicity heavily affects the design of resolution algorithms within the standard Look-Compute-Move (LCM) model. In particular, we have investigated two dense and highly symmetric topologies: complete graphs and complete bipartite graphs. We characterized all solvable configurations for synchronous robots, whereas it is known that in complete graphs asynchronous robots cannot solve the problem, ever. Instead of approaching directly the asynchronous case in complete bipartite graphs, we asked what happens in the so-called semi-synchronous model, that is robots are synchronized but they are not necessarily all active within all LCM cycles. It turns out that still the gathering can never be accomplished on complete graphs, whereas challenging cases arise in complete bipartite graphs. We provide a distributed algorithm solving the problem for a wide set of possible configurations. For most of the remaining ones instead we provide impossibility results and a few of ad hoc resolution algorithms studied for very specific cases. Over all, still a full characterization is missing but our study points out how difficult might be to derive a general argument that catches all peculiarities. Moreover, some of our approaches reveal new insights that might be very useful for the resolution of other tasks.

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

The rainbow number of matchings in regular bipartite graphs