In this paper we determine the maximum genus of a graph by using the matching number of the intersection graph of a basis of its cycle space. Our result is a common generalization of a theorem of Glukhov and a theorem of Nebeský .

Bondy and Erdős [2] have conjectured that the Ramsey number for three cycles Cₖ of odd length has value r(Cₖ,Cₖ,Cₖ) = 4k-3. We give a proof that r(C₇,C₇,C₇) = 25 without using any computer support.

A degree monotone path in a graph G is a path P such that the sequence of degrees of the vertices in the order in which they appear on P is monotonic. The length (number of vertices) of the longest degree monotone path in G is denoted by mp(G). This parameter, inspired by the well-known Erdős- Szekeres theorem, has been studied by the authors in two earlier papers. Here we consider a saturation problem for the parameter mp(G). We call G saturated if, for every edge e added to G, mp(G + e) > mp(G),...

In this paper, the effects on the signless Laplacian spectral radius of a graph are studied when some operations, such as edge moving, edge subdividing, are applied to the graph. Moreover, the largest signless Laplacian spectral radius among the all unicyclic graphs with $n$ vertices and $k$ pendant vertices is identified. Furthermore, we determine the graphs with the largest Laplacian spectral radii among the all unicyclic graphs and bicyclic graphs with $n$ vertices and $k$ pendant vertices, respectively....

We prove several results about the structure of 2-factors in iterated line graphs. Specifically, we give degree conditions on G that ensure L²(G) contains a 2-factor with every possible number of cycles, and we give a sufficient condition for the existence of a 2-factor in L²(G) with all cycle lengths specified. We also give a characterization of the graphs G where $L^k\left(G\right)$ contains a 2-factor.

We give the Turán number ex (n, 2P5) for all positive integers n, improving one of the results of Bushaw and Kettle [Turán numbers of multiple paths and equibipartite forests, Combininatorics, Probability and Computing, 20 (2011) 837-853]. In particular we prove that ex (n, 2P5) = 3n−5 for n ≥ 18.

Let ex (n,G) denote the maximum number of edges in a graph on n vertices which does not contain G as a subgraph. Let Pi denote a path consisting of i vertices and let mPi denote m disjoint copies of Pi. In this paper we count ex(n, 3P4)

A graph $G$ is called $\{H_1,H_2,\cdots ,H_k\}$-free if $G$ contains no induced subgraph isomorphic to any graph $H_i$, $1\le i\le k$. We define $${\sigma }_k=min\left\{\sum _{i=1}^{k}d\left(v_i\right):\{v_1,\cdots ,v_k\}\text{is}\text{an}\text{independent}\text{set}\text{of}\text{vertices}\text{in}G\right\}.$$ In this paper, we prove that (1) if $G$ is a connected $\{K_{1,4},K_{1,4}+e\}$-free graph of order $n$ and ${\sigma }_3\left(G\right)\ge n-1$, then $G$ is traceable, (2) if $G$ is a 2-connected $\{K_{1,4},K_{1,4}+e\}$-free graph of order $n$ and $|N\left(x_1\right)\cup N\left(x_2\right)|+|N\left(y_1\right)\cup N\left(y_2\right)|\ge n-1$ for any two distinct pairs of non-adjacent vertices $\{x_1,x_2\}$, $\{y_1,y_2\}$ of $G$, then $G$ is traceable, i.e., $G$ has a Hamilton path, where $K_{1,4}+e$ is a graph obtained by joining a pair of non-adjacent vertices in a $K_{1,4}$.

In a bidirected graph, an edge has a direction at each end, so bidirected graphs generalize directed graphs. We generalize the definitions of transitive closure and transitive reduction from directed graphs to bidirected graphs by introducing new notions of bipath and bicircuit that generalize directed paths and cycles. We show how transitive reduction is related to transitive closure and to the matroids of the signed graph corresponding to the bidirected graph.