Let G be an edge-colored graph. If every subpath of length at most l + 1 within a path P in a graph G consists of uniquely colored edges, then P is called an l-rainbow path. A connected graph G is deemed ( 1 , 2 )-rainbow connected if there ...

Let t be a positive real number. A graph is called t-tough if the removal of any vertex set S that disconnects the graph leaves at most |S|/t components. The toughness of a graph is the largest t for which the graph is t-tough. We prove that ...$\mathcal{}{}^{\mathrm{}}{}^{\mathrm{}}$$$

Cycles are the only 2-connected graphs in which any two nonadjacent vertices form a vertex cut. We generalize this fact by proving that for every integer $k\ge 3$ there exists a unique graph G satisfying the following three conditions: (1) G is k-...

Given t ≥ 2 and 0 ≤ k ≤ t, we prove that the number of labelled k-connected chordal graphs with n vertices and tree-width at most t is asymptotically c n − 5 / 2 γ n n !, as n → ∞, for some constants c , γ > 0 depending on t and k. Additionally, ...

The Steiner tree problem in graphs is widely studied because of its usefulness in network design and circuit layout. In this context, given a set of vertices $S(|S|\ge 2,)$ a tree that connects all vertices in S is called an S-Steiner tree. This helps ...${}^{}$${}^{}\mathrm{}$${}^{}$${}_{}$$\mathrm{}$${}_{}$${}_{}{}_{}$${}_{\mathrm{}}{}_{}$$\mathrm{}\mathrm{}$${}_{\mathrm{}}$${}_{}$

Let R be a finite commutative ring with identity. We study the structure of the zero-divisor graph of R and then determine its vertex connectivity when: (i) R is a local principal ideal ring, and (ii) R is a finite direct product of local ...

In this paper, we investigate the connectivity of friends-and-strangers graphs, which were introduced by Defant and Kravitz in 2020. We begin by considering friends-and-strangers graphs arising from two random graphs and consider the threshold ...

Let G be a nontrivial connected and vertex-colored graph. A vertex subset X is called rainbow if any two vertices in X have distinct colors. The graph G is called rainbow vertex-disconnected if for any two vertices x and y of G, there exists a ...$$$$$$$$${}_{\mathrm{}}$$\mathrm{}$$\mathrm{}$${}^{\frac{\mathrm{}}{\mathrm{}}}$$$

Identifying certain conditions that ensure the Hamiltonicity of graphs is highly important and valuable due to the fact that determining whether a graph is Hamiltonian is an NP-complete problem.For a graph G with vertex set V(G) and edge set E(G), ...${}_{\mathrm{}}$${}_{\mathrm{}}$${}_{\mathrm{}}\underset{{}_{}{}_{}}{}{}_{}{}_{}{}_{}{}_{}$${}_{\mathrm{}}\underset{{}_{}{}_{}}{}{}_{}{}_{}{}_{}{}_{}$${}_{}{}_{}$${}_{}$$\mathrm{}$$\mathrm{}{}_{\mathrm{}}{}_{\mathrm{}}$$\mathrm{}$

For a graph, its path-index is the greatest eigenvalue of its path-matrix, whose ( x , y )-entry for vertices x and y equals the connectivity of x and y if x ≠ y and 0 otherwise. Some upper and lower bounds are established on the path-...

• The path-index is estimated.
• Tight upper and lower bounds on the path index are ...

For $0\le t\le r$ let m(t, r) be the maximum number s such that every t-edge-connected r-graph has s pairwise disjoint perfect matchings. There are only a few values of m(t, r) known, for instance $m(3,3)=m(4,r)=1$, and $m(t,r)\le r-2$ for all $t\ne 5$, and $m(t,r)\le r-\mathrm{...}$$\mathrm{}\mathrm{}\mathrm{}$$\mathrm{}$$\mathrm{}$

A graph that can be generated from $K_1$ using joins and 0-sums is called a cograph. We define a sesquicograph to be a graph that can be generated from $K_1$ using joins, 0-sums, and 1-sums. We show that, like cographs, sesquicographs are closed under ...

The (strong) proper vertex connection number s p v c → ( Γ ), ( s ) p v c-number for short, is denoted as the smallest cardinality of colors required to color the digraph Γ so that Γ is (strong) properly vertex connected. The pvc-number and the s ...

• We present some fundamental results on the proper vertex connection number and the strong proper vertex connection number of digraphs.
• We determine the proper vertex connection number and the strong proper vertex connection number for ...

The generalized k-connectivity κ k ( G ) of a graph G, introduced by Hager in 1985, is a natural generalization of the concept of connectivity κ ( G ), which is just for k = 2. Rongxia Hao et al. determined the generalized 4-connectivity of the ...

An edge-colored graph is called rainbow if every two edges receive distinct colors, and called proper if every two adjacent edges receive distinct colors. An edge-colored graph $G^c$ is properly vertex/edge-pancyclic if every vertex/edge of the graph ...$\mathrm{}$${}^{}$${}^{}$${}^{}$${}^{}$${}^{}$${}^{}$${}^{}$${}^{}$${}_{}^{}$${}^{}{}_{}\frac{}{\mathrm{}}\mathrm{}$${}^{}{}_{}\frac{}{\mathrm{}}\mathrm{}$${}_{}^{}$${}^{}{}_{}\mathrm{}$${}_{\mathrm{}}$${}_{}^{}$${}^{}{}_{}\mathrm{}$${}_{\mathrm{}}$${}_{\mathrm{}}$

Let G be a graph such that, whenever two vertices x and y of G are joined by three internally disjoint paths, x and y are adjacent. Jamison and Mulder determined that the set of such graphs coincides with the set of graphs that can be ...

Benjamini and Papasoglou (2011) showed that planar graphs with uniform polynomial volume growth admit 1-dimensional annular separators: The vertices at graph distance R from any vertex can be separated from those at distance 2R by removing at most ...$\mathrm{}$$\mathrm{}$$\mathrm{}$$\mathrm{}$${\mathbb{}}^{}$${}^{}$

A graph is called supereulerian if it has a spanning eulerian subgraph. A connected even [2, 2s]-factor of a graph G is a connected factor with all vertices of even degree $i(i\in \{2,4,\dots ,2s\})$, where $s\ge 1$ is an integer. Let $W_{s,t}$ be the graph obtained ...${}_{\mathrm{}}$$\mathrm{}{}_{\mathrm{}}$${}_{\mathrm{}}$${}_{\mathrm{}}$${}_{\mathrm{}}$$\mathrm{}$${}_{\mathrm{}\mathrm{}}$${}_{\mathrm{}}$$\mathrm{}$$\mathrm{}\mathrm{}$$\mathcal{}$$\mathcal{}$$$\begin{array}{c}\hfill \mathcal{}{}_{\mathrm{}\mathrm{}\mathrm{}}{}_{\mathrm{}}{}_{\mathrm{}}{}_{\mathrm{}\mathrm{}}{}_{\mathrm{}}{}_{\mathrm{}\mathrm{}}\end{array}$$$\mathrm{}\mathrm{}$$\mathrm{}\mathrm{}$$\mathcal{}{}_{\mathrm{}\mathrm{}\mathrm{}}{}_{\mathrm{}}{}_{\mathrm{}}{}_{\mathrm{}\mathrm{}}$$\mathcal{}$

We consider the complexity of minmax partitioning of graphs, hypergraphs and (symmetric) submodular functions. Our main result is an algorithm for the problem of partitioning the ground set of a given symmetric submodular function $f:2^V\to \mathbb{R}$ into k ...${}_{\mathrm{}}{}_{\mathrm{}}{}_{}$${}_{\mathrm{}}^{}{}_{}$${}^{{}^{\mathrm{}}}$$$$$$$$$

We present approximation algorithms for several network design problems in the model of flexible graph connectivity (Adjiashvili et al., in: IPCO, pp 13–26, 2020, Math Program 1–33, 2021). Let $k\ge 1$, $p\ge 1$ and $q\ge 0$ be integers. In an instance of the (p,...$$$$$\mathcal{}$$\mathcal{}$${\mathbb{}}_{\mathrm{}}$$$$$${}^{}\mathcal{}$${}^{}$${}^{}$${}_{}{}_{}$$\mathrm{}\mathrm{}$$\mathrm{}\mathrm{}$$\mathrm{}$$\mathrm{}$$\mathrm{}$$$$$$\mathrm{}\mathrm{}$$$$$$\mathrm{}{}_{}$$$${}_{}$