Related papers: Resistors in dual networks
We study the minimum spanning tree problem on the complete graph $K_n$ where an edge $e$ has a weight $W_e$ and a cost $C_e$, each of which is an independent copy of the random variable $U^\gamma$ where $\gamma\leq 1$ and $U$ is the uniform…
The weight of the minimum spanning tree in a complete weighted graph with random edge weights is a well-known problem. For various classes of distributions, it is proved that the weight of the minimum spanning tree tends to a constant,…
Today's communication networks have stringent availability requirements and hence need to rapidly restore connectivity after failures. Modern networks thus implement various forms of fast reroute mechanisms in the data plane, to bridge the…
A multigraph G is triangle decomposable if its edge set can be partitioned into subsets, each of which induces a triangle of G, and rationally triangle decomposable if its triangles can be assigned rational weights such that for each edge e…
For fixed integers $r\ge 3,e\ge 3,v\ge r+1$, an $r$-uniform hypergraph is called $\mathscr{G}_r(v,e)$-free if the union of any $e$ distinct edges contains at least $v+1$ vertices. Brown, Erd\H{o}s and S\'{o}s showed that the maximum number…
A class of graphs is bridge-addable if given a graph $G$ in the class, any graph obtained by adding an edge between two connected components of $G$ is also in the class. We prove a conjecture of McDiarmid, Steger, and Welsh, that says that…
Given a group $G$, the enhanced power graph of $G$ denoted by $\mathcal{G}_e(G)$, is the graph with vertex set $G$ and two distinct vertices $x, y$ are edge connected in $\mathcal{G}_e(G)$ if there exists $z\in G $ such that $x=z^m$ and $…
We consider a cops and robber game where the cops are blocking edges of a graph, while the robber occupies its vertices. At each round of the game, the cops choose some set of edges to block and right after the robber is obliged to move to…
Given a graph with edge costs, the {\em power} of a node is themaximum cost of an edge incident to it, and the power of a graph is the sum of the powers of its nodes. Motivated by applications in wireless networks, we consider the following…
A graph $G$ is called a pairwise compatibility graph (PCG) if there exists an edge-weighted tree $T$ and two non-negative real numbers $d_{min}$ and $d_{max}$ such that each leaf $l_u$ of $T$ corresponds to a vertex $u \in V$ and there is…
Let $G$ be a complete multi-partite graph of order $n$. In this paper, we consider the anti-Ramsey number $ar(G,\mathcal{T}_{q})$ with respect to $G$ and the set $\mathcal{T}_{q}$ of trees with $q$ edges, where $2\le q\le n-1$. For the case…
Effective resistance (ER) is an attractive way to interrogate the structure of graphs. It is an alternative to computing the eigenvectors of the graph Laplacian. One attractive application of ER is to point clouds, i.e. graphs whose…
Given a graph $G=(V,E)$ on $n$ vertices and an assignment of colours to its edges, a set of edges $S \subseteq E$ is said to be rainbow if edges from $S$ have pairwise different colours assigned to them. In this paper, we investigate…
In recent years, the notion of r-robustness for the communication graph of the network has been introduced to address the challenge of achieving consensus in the presence of misbehaving agents. Higher r-robustness typically implies higher…
We provide a structural description of, and invariants for, maximum spanning tree-packable graphs, i.e. those graphs G for which the edge connectivity of G is equal to the maximum number of edge-disjoint spanning trees in G. These graphs…
Graph connectivity is a fundamental combinatorial optimization problem that arises in many practical applications, where usually a spanning subgraph of a network is used for its operation. However, in the real world, links may fail…
For a graph $G$ and $E \subseteq E(G)$, $E$-restricted strong trace is a closed walk which traverses every edge from $E$ once in each direction and every other edge twice in the same direction. In addition, every time a strong trace come to…
A subgraph $H$ of an edge-colored graph $G$ is rainbow if all the edges of $H$ receive different colors. If $G$ does not contain a rainbow subgraph isomorphic to $H$, we say that $G$ is rainbow $H$-free. For connected graphs $H_1$ and…
For every $r\in \mathbb{N}$, we denote by $\theta_{r}$ the multigraph with two vertices and $r$ parallel edges. Given a graph $G$, we say that a subgraph $H$ of $G$ is a model of $\theta_{r}$ in $G$ if $H$ contains $\theta_{r}$ as a…
In this work, we study the color discrepancy of spanning trees in random graphs. We show that for the Erd\H{o}s-R\'enyi random graph $G(n,p)$ with $p$ above the connectivity threshold, the following holds with high probability: in every…