相关论文: Additive bases and flows in graphs

A graph property (i.e., a set of graphs) is induced-hereditary or additive if it is closed under taking induced-subgraphs or disjoint unions. If $\cP$ and $\cQ$ are properties, the product $\cP \circ \cQ$ consists of all graphs $G$ for…

Firstly, for a general graph, we find a recursion formula on the number of Hamiltonian cycles and one on cycles. By this result, we give some new polynomial invariants. Secondly, we give a condition to tell whether a polynomial defined by…

We introduce the Density Formula for (topological) drawings of graphs in the plane or on the sphere, which relates the number of edges, vertices, crossings, and sizes of cells in the drawing. We demonstrate its capability by providing…

Let $G$ be a connected graph on $n$ vertices with adjacency matrix $A_G$. Associated to $G$ is a polynomial $d_G(x_1,\dots, x_n)$ of degree $n$ in $n$ variables, obtained as the determinant of the matrix $M_G(x_1,\dots,x_n)$, where…

Given a constant $\alpha>0$, an $n$-vertex graph is called an $\alpha$-expander if every set $X$ of at most $n/2$ vertices in $G$ has an external neighborhood of size at least $\alpha|X|$. Addressing a question posed by Friedman and…

Recent advances in dynamic graph processing have enabled the analysis of highly dynamic graphs with change at rates as high as millions of edge changes per second. Solutions in this domain, however, have been demonstrated only for…

Markoff mod-$p$ graphs are conjectured to be connected for all primes $p$. In this paper, we use results of Chen and Bourgain, Gamburd, and Sarnak to confirm the conjecture for all $p > 3.448\cdot10^{392}$. We also provide a method that…

Consider the undirected graph $G_n=(V_n, E_n)$ where $V_n = (Z/nZ)^2$ and $E_n$ contains an edge from $(x,y)$ to $(x+1,y)$, $(x,y+1)$, $(x+y,y)$, and $(x,y+x)$ for every $(x,y) \in V_n$. Gabber and Galil, following Margulis, gave an…

A graph is near-planar if it can be obtained from a planar graph by adding an edge. We show the surprising fact that it is NP-hard to compute the crossing number of near-planar graphs. A graph is 1-planar if it has a drawing where every…

An n-vertex graph is called C-Ramsey if it has no clique or independent set of size C log n. All known constructions of Ramsey graphs involve randomness in an essential way, and there is an ongoing line of research towards showing that in…

A consistent path system in a graph $G$ is an intersection-closed collection of paths, with exactly one path between any two vertices in $G$. We call $G$ metrizable if every consistent path system in it is the system of geodesic paths…

A connected graph $\Gamma=(V,E)$ of valency at least $3$ is called a basic $2$-arc-transitive graph if its full automorphism group has a subgroup $G$ with the following properties: (i) $G$ acts transitively on the set of $2$-arcs of…

A cyclic base ordering of a connected graph $G$, is a cyclic ordering of $E(G)$ such that every cyclically consecutive $|V(G)|-1$ edges form a spanning tree. In this project, we study cyclic base ordering of various families of graphs,…

Joint degree vectors give the number of edges between vertices of degree $i$ and degree $j$ for $1\le i\le j\le n-1$ in an $n$-vertex graph. We find lower and upper bounds for the maximum number of nonzero elements in a joint degree vector…

In 1984, Erd\H{o}s and Simonovits conjectured the following: given a bipartite graph $H$, there exist constants $\beta, C > 0$ such that any graph $G$ on $n$ vertices and $pn^2\geq C \mathrm{ex}(n, H)$ edges contains at least $\beta…

In a graph, we assign distinct integers to the vertices, and take the sum of two integers if they are on two adjacent vertices. The minimum possible number of different sums is the \emph{sum index} of this graph. In this paper, we present…

Let $e$ be a positive integer, $p$ be an odd prime, $q=p^{e}$, and $\Bbb F_q$ be the finite field of $q$ elements. Let $f,g \in \Bbb F_q [X,Y]$. The graph $G=G_q(f,g)$ is a bipartite graph with vertex partitions $P=\Bbb F_q^3$ and $L=\Bbb…

Reed and Wood and independently Norine, Seymour, Thomas, and Wollan proved that for each positive integer $t$ there is a constant $c(t)$ such that every graph on $n$ vertices with no $K_t$-minor has at most $c(t)n$ cliques. Wood asked in…

We generalise to signed graphs a classical result of Tutte [Canad. J. Math. 8 (1956), 13--28] stating that every integer flow can be expressed as a sum of characteristic flows of circuits. In our generalisation, the r\^ole of circuits is…

A graph G is called (2k, k)-connected if G is 2k-edge-connected and G-v is k-edge-connected for every vertex v. The study of (2k, k)-connected graphs is motivated by a conjecture of Frank which states that a graph has a 2-vertex-connected…