If in a directed graph, v is an out-neighbor of u and w is an out-neighbor of v but not of u, then w is said to be a second out-neighbor of u. A vertex in a directed graph is said to have a large second neighborhood if it has at least as many second out-neighbors as out-neighbors. The Second Neighborhood Conjecture, first stated by Seymour, asserts that there is a vertex having a large second neighborhood in every oriented graph (a directed graph without loops or digons). It is straightforward to see that the conjecture is true for any oriented graph whose underlying undirected graph is bipartite. We extend this to show that the conjecture holds for oriented graphs whose vertex set can be partitioned into an independent set and a 2-degenerate graph. Fisher proved the conjecture for tournaments and later Havet and Thomassé provided a different proof for the same using median orders of tournaments. Havet and Thomassé in fact showed the stronger statement that if a tournament contains no sink, then it contains at least two vertices with large second neighborhoods. Using their techniques, Fidler and Yuster showed that the conjecture remains true for tournaments from which either a matching or a star has been removed. We extend this result to show that the conjecture holds even for tournaments from which both a matching and a star have been removed. This implies that a tournament from which a matching has been removed contains either a sink or two vertices with large second neighborhoods.

The axiomatic study on the interval function, induced path function and all-paths function of a connected graph is a well-known area in metric graph theory and related areas. In this paper, we introduce the following new axiom: (cp) v is an element of R(u, w) and v is an element of R(u, x) double right arrow w is an element of R(v, x) or x is an element of R(v, w), for all distinct u, v, w, x is an element of V. We present characterizations of (claw, paw)-free graphs using axiom (cp) on the standard path transit functions on graphs, namely the interval function, the induced path function, and the all-paths function. We study the underlying graphs of the transit functions which are (claw, paw)-free and Hamiltonian. We present an axiomatic characterization of the interval function on (claw, paw)-free graphs. Furthermore, we obtain an axiomatic characterization of the induced path function on a subclass of (claw, paw)-free graphs. (C) 2018 Elsevier B.V. All rights reserved.

We prove that the regularity of binomial edge ideals of graphs obtained by gluing two graphs at a free vertex is the sum of the regularity of individual graphs. As a consequence, we generalize certain results of Zafar and Zahid (Electron J Comb 20(4), 2013). We obtain an improved lower bound for the regularity of trees. Further, we characterize trees which attain the lower bound. We prove an upper bound for the regularity of certain subclass of block-graphs. As a consequence, we obtain sharp upper and lower bounds for a class of trees called lobsters.

In this article, we obtain an improved upper bound for the regularity of binomial edge ideals of trees.

In this note we extend the Mulder-Nebeský characterization of the interval function of a connected graph to the disconnected case. One axiom needs to be adapted, but also a new axiom is needed in addition.

Let G be a finite simple graph and I(G) denote the corresponding edge ideal. For all \(s \ge 1\), we obtain upper bounds for \({\text {reg}}(I(G)^s)\) for bipartite graphs. We then compare the properties of G and \(G'\), where \(G'\) is the graph associated with the polarization of the ideal \((I(G)^{s+1} : e_1\cdots e_s)\), where \(e_1,\cdots , e_s\) are edges of G. Using these results, we explicitly compute \({\text {reg}}(I(G)^s)\) for several subclasses of bipartite graphs.

Given $k\ge 1$, a $k$-proper partition of a graph $G$ is a partition ${\mathcal P}$ of $V(G)$ such that each part $P$ of ${\mathcal P}$ induces a $k$-connected subgraph of $G$. We prove that if $G$ is a graph of order $n$ such that $\delta(G)\ge \sqrt{n}$, then $G$ has a $2$-proper partition with at most $n/\delta(G)$ parts. The bounds on the number of parts and the minimum degree are both best possible. We then prove that If $G$ is a graph of order $n$ with minimum degree $\delta(G)\ge\sqrt{c(k-1)n}$, where $c=\frac{2123}{180}$, then $G$ has a $k$-proper partition into at most $\frac{cn}{\delta(G)}$ parts. This improves a result of Ferrara, Magnant and Wenger [Conditions for Families of Disjoint $k$-connected Subgraphs in a Graph, Discrete Math. 313 (2013), 760--764] and both the degree condition and the number of parts are best possible up to the constant $c$.

We prove that the strong chromatic index of a 2-degenerate graph is linear in the maximum degree . This includes the class of all chordless graphs (graphs in which every cycle is induced) which in turn includes graphs where the cycle lengths are multiples of four, and settles a problem by Faudree et al. (Ars Combin 29(B) (1990), 205211). (c) 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 119126, 2013

We obtain some improved upper and lower bounds on the oriented chromatic number for different classes of products of graphs.

We present some classes of graphs which satisfy the acyclic edge colouring conjecture which states that any graph can be acyclically edge coloured with at most ∆ + 2 colours.

We propose the following problem. For some k 1, a graph G is to be properly edge coloured such that any two adjacent vertices share at most k colours. We call this the k-intersection edge colouring. The minimum number of colours sucien t to guarantee such a colouring is the k-intersection chromatic index and is denoted 0(G). Let fk be dened by fk() = max G:( G)= f 0 k (G)g:

An acyclic edge colouringof a graph is a proper edge colouring such that there are no bic hromatic cycles. Theacyclic chromatic indexof a graph is the minimum number k such that there is an acyclic edge colouring using k colours and it is denoted bya′(G). Here, we obtain tight estimates on a′(G) for nontrivial subclasses of the family of 2-degenerate graphs. Specifically, we obtain values of the acyclic chromatic ind ex for the families ofpartial 2-treesandseries-parallel graphs. Another family contained within the family of 2-degenerat graphs is the family of outerplanar graphs . The acyclic chromatic index for outerplanar graphs has already been tig htly determined. It was conjectured by Alon, Sudakov and Zak s thata′(G) ≤ ∆ + 2, where∆ = ∆(G) denotes the maximum degree of the graph. Here we verify the co nj ture for the classes of graphs considered. We first prove that a′(G) ≤ ∆ + 1 for partial 2-trees. As a corollary, it follows that the same bound holds for the class of series-parallel graphs, as it is a subclass of the class of partial 2-trees. These are best pos sible upper bounds, as there are examples graphs in the correspond ing families which require that many colours for an acyclic edge colouring. All these bounds are proved constructively , l ading to effecient algorithms to produce colourings wit h the stated number of colours.

We propose the following problem. A graph G is to be properly edge coloured such that any two adjacent vertices share at most k colours. We call this the k-intersection colouring. The minimum number of colours required for such a colouring is the kintersection chromatic index and is denoted χk. Let f be defined by fk(∆) = max G:∆(G)=∆ {χk(G)} We show that fk(∆) = Θ( 2 k ). We also discuss some open problems.

We prove that the acyclic chromatic index a′(G)⩽6Δ for all graphs with girth at least 9. We extend the same method to obtain a bound of 4.52Δ with the girth requirement g⩾220. We also obtain a relationship between g and a′(G).

An acyclic edge colouring of a graph is a proper edge colouring having no 2-coloured cycle, that is, a colouring in which the union of any two colour classes forms a linear forest. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge colouring using k colours and is usually denoted by a′(G). Determining a′(G) exactly is a very hard problem (both theoretically and algorithmically) and is not determined even for complete graphs. We show that a′(G) ≤ Δ(G) + 1, if G is an outerplanar graph. This bound is tight within an additive factor of 1 from optimality. Our proof is constructive leading to an O(n logΔ) time algorithm. Here, Δ = Δ(G) denotes the maximum degree of the input graph.

We determine the values of the acyclic chromatic index of a class of graphs referred to as d-dimensional partial tori. These are graphs which can be expressed as the Cartesian product of d graphs each of which is an induced path or cycle. This class includes some known classes of graphs like d-dimensional meshes (hypergrids), hypercubes, tori, etc. Our estimates are exact except when the graph is a product of a path and a number of odd cycles, in which case the estimates differ by an additive factor of at most 1. Our results are also constructive and provide an optimal (or almost optimal) acyclic edge colouring in polynomial time.