arXiv subjects

Reza Naserasr

Publications and source records attributed to Reza Naserasr.

42 records · Page 3Linked to original sources

arXiv2015

A New Approach Towards a Conjecture on Intersecting Three Longest Paths

In 1966, T. Gallai asked whether every connected graph has a vertex that appears in all longest paths. Since then this question has attracted much attention and many work has been done in this topic. One important open question in this area is to ask whether any three longest paths contains a common vertex in a connected graph. It was conjectured that the answer to this question is positive. In this paper, we propose a new approach in view of distances among longest paths in a connected graph, and give a substantial progress towards the conjecture along the idea.

math.CO↗

arXiv2015

Homomorphisms of binary Cayley graphs

A binary Cayley graph is a Cayley graph based on a binary group. In 1982, Payan proved that any non-bipartite binary Cayley graph must contain a generalized Mycielski graph of an odd-cycle, implying that such a graph cannot have chromatic number 3. We strengthen this result first by proving that any non-bipartite binary Cayley graph must contain a projective cube as a subgraph. We further conjecture that any homo- morphism of a non-bipartite binary Cayley graph to a projective cube must be surjective and we prove some special case of this conjecture.

math.CO↗

arXiv2015

Walk-powers and homomorphism bound of planar graphs

As an extension of the Four-Color Theorem it is conjectured that every planar graph of odd-girth at least $2k+1$ admits a homomorphism to $PC_{2k}=(\mathbb{Z}_2^{2k}, \{e_1, e_2, ...,e_{2k}, J\})$ where $e_i$'s are standard basis and $J$ is all 1 vector. Noting that $PC_{2k}$ itself is of odd-girth $2k+1$, in this work we show that if the conjecture is true, then $PC_{2k}$ is an optimal such a graph both with respect to number of vertices and number of edges. The result is obtained using the notion of walk-power of graphs and their clique numbers. An analogous result is proved for bipartite signed planar graphs of unbalanced-girth $2k$. The work is presented on a uniform frame work of planar consistent signed graphs.

math.CO↗

arXiv2012

Identifying codes in line graphs

An identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbours within the code. We study the edge-identifying code problem, i.e. the identifying code problem in line graphs. If $\ID(G)$ denotes the size of a minimum identifying code of an identifiable graph $G$, we show that the usual bound $\ID(G)\ge \lceil\log_2(n+1)\rceil$, where $n$ denotes the order of $G$, can be improved to $Θ(\sqrt{n})$ in the class of line graphs. Moreover, this bound is tight. We also prove that the upper bound $\ID(\mathcal{L}(G))\leq 2|V(G)|-5$, where $\mathcal{L}(G)$ is the line graph of $G$, holds (with two exceptions). This implies that a conjecture of R. Klasing, A. Kosowski, A. Raspaud and the first author holds for a subclass of line graphs. Finally, we show that the edge-identifying code problem is NP-complete, even for the class of planar bipartite graphs of maximum degree~3 and arbitrarily large girth.

math.CO↗

arXiv2012

Characterizing extremal digraphs for identifying codes and extremal cases of Bondy's theorem on induced subsets

An identifying code of a (di)graph $G$ is a dominating subset $C$ of the vertices of $G$ such that all distinct vertices of $G$ have distinct (in)neighbourhoods within $C$. In this paper, we classify all finite digraphs which only admit their whole vertex set in any identifying code. We also classify all such infinite oriented graphs. Furthermore, by relating this concept to a well known theorem of A. Bondy on set systems we classify the extremal cases for this theorem.

cs.DM↗

arXiv2011

Extremal graphs for the identifying code problem

An identifying code of a graph G is a dominating set C such that every vertex x of G is distinguished from all other vertices by the set of vertices in C that are at distance at most 1 from x. The problem of finding an identifying code of minimum possible size turned out to be a challenging problem. It was proved by N. Bertrand that if a graph on n vertices with at least one edge admits an identifying code, then a minimum identifying code has size at most n-1. Some classes of graphs whose smallest identifying code is of size n-1 were already known, and few conjectures were formulated to classify all these graphs. In this paper, disproving these conjectures, we classify all finite graphs for which all but one of the vertices are needed to form an identifying code. We also classify all infinite graphs needing the whole set of vertices in any identifying code. New upper bounds in terms of the number of vertices and the maximum degree of a graph are also provided.

cs.DM↗