Lehel conjectured that for all n, any 2-edge-coloring of Kn admits a partition of the vertex set into a red cycle and a blue cycle. This conjecture led to a significant amount of work on related questions and was eventually proven for all n by Bessy and Thomassé. Balogh, Barát, Gerbner, Gyárfás, and Sárközy conjectured a stronger statement for large n: that if δ(G) > 3n/4, then any 2-edge-color...

Let the graph G = (V,E) be a cycle with n + 1 vertices, nonnegative vertex weights and positive edge lengths. The inverse 1-median problem on a cycle consists in changing the vertex weights at minimum cost such that a prespecified vertex becomes the 1-median. The cost is proportional to the increase or decrease of the corresponding weight. We show that this problem can be formulated as a linear...

Paired de Bruijn graphs are a variant of classic de Bruijn graphs used in genome assembly. In these graphs, each vertex v is associated with two labels L(v) and R(v). We study the NP-hard Sound Covering Cycle problem which has as input a paired de Bruijn graph G and two integers d and `, and the task is to find a length-` cycle C containing all arcs of G such that for every vertex v in C and th...

A graph dynamical system consists of an undirected graph Y where each vertex is assigned a state from a finite set K, an update function for each vertex that depends only on the state of the vertex and its neighbors, and a method that dictates how to compose these functions. A common subclass of graph dynamical systems are sequential dynamical systems, or SDSs, where the update functions are ap...

A tight k-uniform `-cycle, denoted by TC ` , is a k-uniform hypergraph whose vertex set is v0, · · · , v`−1, and the edges are all the k-tuples {vi, vi+1, · · · , vi+k−1}, with subscripts modulo k. Motivated by a classic result in graph theory that every n-vertex cycle-free graph has at most n− 1 edges, Sós and, independently, Verstraëte asked whether for every integer k, a k-uniform n-vertex h...

a emph{signed graph} (or, in short, emph{sigraph}) $s=(s^u,sigma)$ consists of an underlying graph $s^u :=g=(v,e)$ and a function $sigma:e(s^u)longrightarrow {+,-}$, called the signature of $s$. a emph{marking} of $s$ is a function $mu:v(s)longrightarrow {+,-}$. the emph{canonical marking} of a signed graph $s$, denoted $mu_sigma$, is given as $$mu_sigma(v) := prod_{vwin e(s)}sigma(vw).$$the li...

Rainbow vertex-connection number is the minimum colors assignment to vertices of graph, such that each vertex connected by a path whose edges have distinct and denoted . The rainbow connection can be applied graphs resulting from operations. One methods create new graph perform operations between two graphs. Thus, this research uses comb product operation determine rainbow-vertex cycle complete...

The line graph of an edge-signed graph carries vertex signs. A vertex-signed graph is consistent if every circle (cycle, circuit) has positive vertex-sign product. Acharya, Acharya, and Sinha recently characterized edge-signed graphs whose line graphs are consistent. Their proof applies Hoede’s relatively difficult characterization of consistent vertex-signed graphs. We give a different, constr...

A chordless cycle in a graph G is an induced subgraph of G which is a cycle of length at least four. We prove that the Erdős-Pósa property holds for chordless cycles, which resolves the major open question concerning the Erdős-Pósa property. Our proof for chordless cycles is constructive: in polynomial time, one can find either k + 1 vertex-disjoint chordless cycles, or ck 2 log k vertices hitt...

In this paper, we prove that a 2-connected weighted graph G contains either a Hamilton cycle or a cycle of weight at least 2m/3 if it satisfies the following conditions: (1) ∑3 i=1 d (vi) ≥ m, where v1, v2 and v3 are three pairwise nonadjacent vertices of G, and two of them are nonadjacent vertices of an induced claw or an induced modified claw; (2) In each induced claw and each induced modifie...