Star Forest Research Articles

On a conjecture of Erdős on size Ramsey number of star forests

On a conjecture of Erdős on size Ramsey number of star forests

Spectral extremal graphs for edge blow-up of star forests

The edge blow-up of a graph G, denoted by Gp+1, is obtained by replacing each edge of G with a clique of order p+1, where the new vertices of the cliques are all distinct. Yuan (2022) [20] determined the range of the Turán numbers for edge blow-up of all bipartite graphs and the exact Turán numbers for edge blow-up of all non-bipartite graphs. In this paper, we prove that the graphs with the maximum spectral radius in an n-vertex graph without any copy of edge blow-up of star forests are the extremal graphs for edge blow-up of star forests when n is sufficiently large.

Infinite Ramsey-Minimal Graphs for Star Forests

Infinite Ramsey-Minimal Graphs for Star Forests

Antimagic labeling for unions of graphs with many three-paths

Let G be a graph with m edges and let f be a bijection from E(G) to {1,2,…,m}. For any vertex v, denote by ϕf(v) the sum of f(e) over all edges e incident to v. If ϕf(v)≠ϕf(u) holds for any two distinct vertices u and v, then f is called an antimagic labeling of G. We call G antimagic if such a labeling exists. Hartsfield and Ringel [10] conjectured that all connected graphs except P2 are antimagic. Denote the disjoint union of graphs G and H by G∪H, and the disjoint union of t copies of G by tG. For an antimagic graph G (connected or disconnected), we define the parameter τ(G) to be the maximum integer such that G∪tP3 is antimagic for all t⩽τ(G). Chang, Chen, Li, and Pan showed that for all antimagic graphs G, τ(G) is finite [3]. Further, Shang, Lin, Liaw [19] and Li [14] found the exact value of τ(G) for special families of graphs: star forests and balanced double stars, respectively. They did this by finding explicit antimagic labelings of G∪tP3 and proving a tight upper bound on τ(G) for these special families. In the present paper, we generalize their results by proving an upper bound on τ(G) for all graphs. For star forests and balanced double stars, this general bound is equivalent to the bounds given in [19] and [14] and tight. In addition, we prove that the general bound is also tight for every other graph we have studied, including an infinite family of jellyfish graphs, cycles Cn with 3⩽n⩽9, and the double triangle 2C3.

Cooperative Colorings of Forests

Given a family $\mathcal G$ of graphs spanning a common vertex set $V$, a cooperative coloring of $\mathcal G$ is a collection of one independent set from each graph $G \in \mathcal G$ such that the union of these independent sets equals $V$. We prove that for large $d$, there exists a family $\mathcal G$ of $(1+o(1)) \frac{\log d}{\log \log d}$ forests of maximum degree $d$ that admits no cooperative coloring, which significantly improves a result of Aharoni, Berger, Chudnovsky, Havet, and Jiang (Electronic Journal of Combinatorics, 2020). Our family $\mathcal G$ consists entirely of star forests, and we show that this value for $|\mathcal G|$ is asymptotically best possible in the case that $\mathcal G$ is a family of star forests.

Turán problems for edge-ordered graphs

In this paper we initiate a systematic study of the Turán problem for edge-ordered graphs. A simple graph is called edge-ordered if its edges are linearly ordered. This notion allows us to study graphs (and in particular their maximum number of edges) when a subgraph is forbidden with a specific edge-order but the same underlying graph may appear with a different edge-order.We prove an Erdős-Stone-Simonovits-type theorem for edge-ordered graphs—we identify the relevant parameter for the Turán number of an edge-ordered graph and call it the order chromatic number. We establish several important properties of this parameter.We also study Turán numbers of edge-ordered paths, star forests and the cycle of length four. We make strong connections to Davenport-Schinzel theory, the theory of forbidden submatrices, and show an application in discrete geometry.

Classes of graphs without star forests and related graphs

This work provides a structural characterisation of hereditary graph classes that do not contain a star forest, several graphs obtained from star forests by subset complementation, a union of cliques, and the complement of a union of cliques as induced subgraphs. This provides, for instance, structural results for graph classes not containing a matching and several complements of a matching. In terms of the speed of hereditary graph classes, our results imply that all such classes have at most factorial speed of growth.

Acyclic, Star, and Injective Colouring: Bounding the Diameter

We examine the effect of bounding the diameter for a number of natural and well-studied variants of the COLOURING problem. A colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. The corresponding decision problems are ACYCLIC COLOURING, STAR COLOURING and INJECTIVE COLOURING. The last problem is also known as $L(1,1)$-LABELLING and we also consider the framework of $L(a,b)$-LABELLING. We prove a number of (almost-)complete complexity classifications. In particular, we show that for graphs of diameter at most~$d$, ACYCLIC $3$-COLOURING is polynomial-time solvable if $d\leq 2$ but NP-complete if $d\geq 4$, and STAR $3$-COLOURING is polynomial-time solvable if $d\leq 3$ but NP-complete for $d\geq 8.$ As far as we are aware, STAR $3$-COLOURING is the first problem that exhibits a complexity jump for some $d\geq 3.$ Our third main result is that $L(1,2)$-LABELLING is NP-complete for graphs of diameter~$2$; we relate the latter problem to a special case of HAMILTONIAN PATH.

A Note on the Turán Number of an Arbitrary Star Forest

A Note on the Turán Number of an Arbitrary Star Forest

Turán numbers for hypergraph star forests

Fix a graph F. We say that a graph is F-free if it does not contain F as a subhypergraph. The Turán number of F, denoted ex(n,F), is the maximum number of edges possible in an n-vertex F-free graph. The study of Turán numbers is a central problem in graph theory. The goal of this paper is to generalize a theorem of Lidický, Liu and Palmer [Electron. J. of Combin 20 (2016)] that determines ex(n,F) for F a forest of stars. In particular, we consider generalizations of the problem to three different well-studied hypergraph settings and in each case we prove an asymptotic result for all reasonable parameters defining our “star forests”.

Spectral extremal results on the $\alpha$-index of graphs without minors and star forests

Spectral extremal results on the $\alpha$-index of graphs without minors and star forests

Antimagic labeling of forests with sets of consecutive integers

Let G be a graph. We say G is k-shifted antimagic if there exists a bijection f from E(G) to {k+1,…,k+|E(G)|} such that the vertex sum ϕf(v)=∑e∈E(v)f(e) of every vertex v∈V(G) is unique, and is absolutely antimagic if it is k-shifted antimagic for any k∈Z. The ordinary antimagic problem (k=0) of connected graphs was proposed by Hartsfield and Ringel (1990).In this paper, we prove that the P2,P3,P4-free linear forests and the S2-free star forests are absolutely antimagic with only a few exceptions. This extends the results in (Shang, 2016: Shang et al., 2015). Moreover, we prove that the odd tree forests are absolutely antimagic.

Some new results on the Turán number of star forests

Let ex(n,G)=max{e(H)|Hisagraphwith|V(H)|=nandHcontainsnoG}, the Turán number of a graph G, where e(H)=|E(H)|. We determine ex(n,Sℓ1∪Sℓ2) for ℓ2≥3,ℓ1≥2ℓ2+2 and n≥2ℓ1+2ℓ2, where Sℓ is the star of order ℓ+1 and ∪ is the disjoint union. Moreover, we also determine ex(n,Sℓ1∪Sℓ2∪Sℓ3) for ℓ1≥ℓ2≥ℓ3≥1 and n≥max{M1,M2}, where both M1 and M2 are two parameters only depending on ℓ1,ℓ2 and ℓ3. This improves the corresponding results due to Lidický et al. (2013).

On the spectral radius of graphs without a star forest

Let F=∪i=1kSdi be the union of pairwise vertex-disjoint k stars of order d1+1,…,dk+1, respectively, where k≥2 and d1≥⋯≥dk≥1. In this paper, we present two sharp upper bounds for the spectral radius of F-free (bipartite) graphs and characterize all corresponding extremal graphs. Moreover, the minimum least eigenvalue of the adjacency matrix of an F-free graph and all extremal graphs are obtained.

On the anti-Ramsey number of forests

We call a subgraph of an edge-colored graph rainbow, if all of its edges have different colors. The anti-Ramsey number of a graph G in a complete graph Kn, denoted by ar(Kn,G), is the maximum number of colors in an edge-coloring of Kn with no rainbow copy of G. In this paper, we determine the exact value of the anti-Ramsey number for star forests and double stars and the approximate value of the anti-Ramsey number for linear forests.

Minimal ordered Ramsey graphs

An ordered graph is a graph equipped with a linear ordering of its vertex set. A pair of ordered graphs is Ramsey finite if it has only finitely many minimal ordered Ramsey graphs and Ramsey infinite otherwise. Here an ordered graph F is an ordered Ramsey graph of a pair (H,H′) of ordered graphs if for any coloring of the edges of F in colors red and blue there is either a copy of H with all edges colored red or a copy of H′ with all edges colored blue (where copies are not necessarily induced subgraphs). Such an ordered Ramsey graph is minimal if neither of its proper subgraphs is an ordered Ramsey graph of (H,H′). If H=H′ then H itself is called Ramsey finite.We show that a connected ordered graph is Ramsey finite if and only if it is a star with center being the first or the last vertex in the linear order. In general we prove that each Ramsey finite (not necessarily connected) ordered graph H has a pseudoforest as a Ramsey graph and therefore is a star forest with strong restrictions on the positions of the centers of the stars. In the asymmetric case we show that (H,H′) is Ramsey finite whenever H is a so-called monotone matching. Among several further results we show that there are Ramsey finite pairs of ordered stars and ordered caterpillars of arbitrary size and diameter. This is in contrast to the unordered setting where for any Ramsey finite pair (H,H′) of forests either one of H or H′ is a matching or both are star forests (with additional constraints).Several of our results give a relation between Ramsey finiteness and the existence of sparse ordered Ramsey graphs. Motivated by these relations we characterize all pairs of ordered graphs that have a forest as an ordered Ramsey graph and all pairs of ordered graphs that have a pseudoforest as a Ramsey graph.Our results show similarities between the ordered and the unordered setting for graphs containing cycles and significant differences for forests.

On Size Bipartite and Tripartite Ramsey Numbers for The Star Forest and Path on 3 Vertices

For simple graphs G and H the size multipartite Ramsey number mj ( G , H ) is the smallest natural number t such that any arbitrary red-blue coloring on the edges of Kjxt contains a red G or a blue H as a subgraph. We studied the size tripartite Ramsey numbers m 3( G , H ) where G=mK1,n and H=P3 . In this paper, we generalize this result. We determine m3(G,H) where G is a star forest, namely a disjoint union of heterogeneous stars, and H=P3 . Moreover, we also determine m2(G,H) for this pair of graphs G and H .

The Turán number of spanning star forests

The Turán number of spanning star forests

Two results about the Turán number of star forests

The Turán number of a graph H, denoted by ex(n,H), is the maximum number of edges of an n-vertex simple graph having no H as a subgraph. Let Sℓ denote the star on ℓ+1 vertices, and let k⋅Sℓ denote k disjoint copies of Sℓ. Erdős and Gallai determined the value ex(n,k⋅S1) for all positive integers k and n. Yuan and Zhang determined the value ex(n,k⋅S2) and characterized all extremal graphs for all positive integers k and n. Recently, Lan et al. determined the value ex(n,2⋅S3) for all positive integers n. In this paper, we first determine the Turán number ex(n,2⋅Sℓ) for all positive integers ℓ(≥4) and n, and then determine the Turán number ex(n,3⋅Sℓ) for all positive integers ℓ(≥3) and n, improving two of the results of Lidický et al.

On the star forest polytope for trees and cycles

Let G = (V, E) be an undirected graph where the edges in E have non-negative weights. A star in G is either a single node of G or a subgraph of G where all the edges share one common end-node. A star forest is a collection of vertex-disjoint stars in G. The weight of a star forest is the sum of the weights of its edges. This paper deals with the problem of finding a Maximum Weight Spanning Star Forest (MWSFP) in G. This problem is NP-hard but can be solved in polynomial time when G is a cactus [Nguyen, Discrete Math. Algorithms App. 7 (2015) 1550018]. In this paper, we present a polyhedral investigation of the MWSFP. More precisely, we study the facial structure of the star forest polytope, denoted by SFP(G), which is the convex hull of the incidence vectors of the star forests of G. First, we prove several basic properties of SFP(G) and propose an integer programming formulation for MWSFP. Then, we give a class of facet-defining inequalities, called M-tree inequalities, for SFP(G). We show that for the case when G is a tree, the M-tree and the nonnegativity inequalities give a complete characterization of SFP(G). Finally, based on the description of the dominating set polytope on cycles given by Bouchakour et al. [Eur. J. Combin. 29 (2008) 652–661], we give a complete linear description of SFP(G) when G is a cycle.