Free Graphs Research Articles

The χ‐Boundedness of (P2 ∪ P3)‐Free Graphs

In the early 1980s, Gyárfás introduced the concept of the χ‐bound with χ‐binding functions thereby extending the notion of perfectness. There are a number of challenging conjectures about the χ‐bound. Let χ(G), ω(G), and Δ(G) be the chromatic number, clique number, and maximum degree of a graph G, respectively. In this paper, we prove that if G is a triangle‐free and (P2 ∪ P3)‐free graph, then χ(G) ≤ 3 unless G is one of eight graphs with Δ(G) = 5 and χ(G) = 4, where the eight graphs are extended from the Grötzsch graph as a Mycielskian of a 5‐cycle graph. Moreover, we also show that χ(G) ≤ 3ω(G) if G is a {P2 ∪ P3, W4}‐free graph.

Which graphs can be counted in $C_4$-free graphs?

Which graphs can be counted in $C_4$-free graphs?

On the Parameterized Complexity Of the Acyclic Matching Problem

A matching is a set of edges in a graph with no common endpoint. A matching $M$ is called acyclic if the induced subgraph on the endpoints of the edges in $M$ is acyclic. Given a graph $G$ and an integer $k$, Acyclic Matching Problem seeks for an acyclic matching of size $k$ in $G$. The problem is known to be NP-complete. In this paper, we investigate the complexity of the problem in different aspects. First, we prove that the problem remains NP-complete for the class of planar bipartite graphs of maximum degree three and arbitrarily large girth. Also, the problem remains NP-complete for the class of planar line graphs with maximum degree four. Moreover, we study the parameterized complexity of the problem. In particular, we prove that the problem is W[1]-hard on bipartite graphs with respect to the parameter $k$. On the other hand, the problem is fixed parameter tractable with respect to $k$, for line graphs, $C_4$-free graphs and every proper minor-closed class of graphs (including bounded tree-width and planar graphs).

Cocycle superrigidity from higher rank lattices to $ {{\rm{Out}}}{(F_N)} $

<p style='text-indent:20px;'>We prove a rigidity result for cocycles from higher rank lattices to <inline-formula><tex-math id="M2">\begin{document}$ \mathrm{Out}(F_N) $\end{document}</tex-math></inline-formula> and more generally to the outer automorphism group of a torsion-free hyperbolic group. More precisely, let <inline-formula><tex-math id="M3">\begin{document}$ G $\end{document}</tex-math></inline-formula> be either a product of connected higher rank simple algebraic groups over local fields, or a lattice in such a product. Let <inline-formula><tex-math id="M4">\begin{document}$ G \curvearrowright X $\end{document}</tex-math></inline-formula> be an ergodic measure-preserving action on a standard probability space, and let <inline-formula><tex-math id="M5">\begin{document}$ H $\end{document}</tex-math></inline-formula> be a torsion-free hyperbolic group. We prove that every Borel cocycle <inline-formula><tex-math id="M6">\begin{document}$ G\times X\to \mathrm{Out}(H) $\end{document}</tex-math></inline-formula> is cohomologous to a cocycle with values in a finite subgroup of <inline-formula><tex-math id="M7">\begin{document}$ \mathrm{Out}(H) $\end{document}</tex-math></inline-formula>. This provides a dynamical version of theorems of Farb–Kaimanovich–Masur and Bridson–Wade asserting that every homomorphism from <inline-formula><tex-math id="M8">\begin{document}$ G $\end{document}</tex-math></inline-formula> to either the mapping class group of a finite-type surface or the outer automorphism group of a free group, has finite image.</p><p style='text-indent:20px;'>The main new geometric tool is a barycenter map that associates to every triple of points in the boundary of the (relative) free factor graph a finite set of (relative) free splittings.</p>

On [formula omitted]-hued list coloring of [formula omitted]-minor free graphs

For a given list assignment L of a graph G, an (L,r)-coloring of G is a proper coloring c such that for any vertex v with degree d(v), v is adjacent to vertices of at least min{d(v),r} different color with c(v)∈L(v). The r-hued list chromatic number of G, denoted as χL,r(G), is the least integer k, such that for any v∈V(G) and every list assignment L with |L(v)|=k, G has an (L,r)-coloring. Let K(r)=r+3 if 2≤r≤3, K(r)=⌊3r/2⌋+1 if r≥4. In Song et al. (2014), it is proved that if G is a K4-minor-free graph, then χL,r(G)≤K(r)+1. Let K4(n) be the set of all subdivisions of K4 on n vertices. Utilizing the decompositions by Chen et al for K4(7)-minor free graphs in Chen et al. (2020), we prove that if G is a K4(7)-minor free graph, then χL,r(G)≤K(r)+1.

Computing subset transversals in H-free graphs

We study the computational complexity of two well-known graph transversal problems, namely Subset Feedback Vertex Set and Subset Odd Cycle Transversal, by restricting the input to H-free graphs, that is, to graphs that do not contain some fixed graph H as an induced subgraph. By combining known and new results, we determine the computational complexity of both problems on H-free graphs for every graph H except when H=sP1+P4 for some s≥1. As part of our approach, we introduce the Subset Vertex Cover problem and prove that it is polynomial-time solvable for (sP1+P4)-free graphs for every s≥1.

Upper paired domination versus upper domination

A paired dominating set $P$ is a dominating set with the additional property that $P$ has a perfect matching. While the maximum cardainality of a minimal dominating set in a graph $G$ is called the upper domination number of $G$, denoted by $\Gamma(G)$, the maximum cardinality of a minimal paired dominating set in $G$ is called the upper paired domination number of $G$, denoted by $\Gamma_{pr}(G)$. By Henning and Pradhan (2019), we know that $\Gamma_{pr}(G)\leq 2\Gamma(G)$ for any graph $G$ without isolated vertices. We focus on the graphs satisfying the equality $\Gamma_{pr}(G)= 2\Gamma(G)$. We give characterizations for two special graph classes: bipartite and unicyclic graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ by using the results of Ulatowski (2015). Besides, we study the graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and a restricted girth. In this context, we provide two characterizations: one for graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and girth at least 6 and the other for $C_3$-free cactus graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$. We also pose the characterization of the general case of $C_3$-free graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ as an open question.

Linearly free graphs

AbstractIn this paper we are interested in an intrinsic property of graphs which is derived from their embeddings into the Euclidean 3‐space . An embedding of a graph into is said to be linear, if it sends every edge to be a line segment. And we say that an embedding of a graph into is free, if is a free group. Lastly a simple connected graph is said to be linearly free if every its linear embedding is free. In the 1980s it was proved that every complete graph is linearly free, by Nicholson. In this paper, we develop Nicholson's arguments into a general notion, and establish a sufficient condition for a linear embedding to be free. As an application of the condition we give a partial answer for a question: How much can the complete graph be enlarged so that the linear‐freeness is preserved and the clique number does not increase? And an example supporting our answer is provided. As the second application it is shown that a simple connected graph of minimal valency at least 3 is linearly free, if it has less than eight vertices. The conditional inequality is strict, because we found a graph with eight vertices which are not linearly free. It is also proved that for the complete bipartite graph is linearly free.

Paths of Length Three are $K_{r+1}$-Turán-Good

The generalized Turán problem ex$(n,T,F)$ is to determine the maximal number of copies of a graph $T$ that can exist in an $F$-free graph on $n$ vertices. Recently, Gerbner and Palmer noted that the solution to the generalized Turán problem is often the original Turán graph. They gave the name "$F$-Turán-good" to graphs $T$ for which, for large enough $n$, the solution to the generalized Turán problem is realized by a Turán graph. They prove that the path graph on two edges, $P_2$, is $K_{r+1}$-Turán-good for all $r \ge 3$, but they conjecture that the same result should hold for all $P_\ell$. In this paper, using arguments based in flag algebras, we prove that the path on three edges, $P_3$, is also $K_{r+1}$-Turán-good for all $r \ge 3$.

Forbidden pairs of disconnected graphs for 2‐factor of connected graphs

AbstractLet be a set of graphs. A graph is said to be ‐free if does not contain as an induced subgraph for all in , and we call a forbidden pair if . In 2008, Faudree et al. characterized all pairs of connected graphs such that every 2‐connected ‐free graph of sufficiently large order has a 2‐factor. In 2013, Fujisawa et al. characterized all pairs of connected graphs such that every connected ‐free graph of sufficiently large order with minimum degree at least two has a 2‐factor. In this paper, we generalize these two results by considering disconnected graphs . In other words, we characterize all pairs of graphs such that every 2‐connected ‐free graph of sufficiently large order has a 2‐factor. We also characterize all pairs of graphs such that every connected ‐free graph of sufficiently large order with minimum degree at least two has a 2‐factor.

Generalized Planar Turán Numbers

In a generalized Turán problem, we are given graphs $H$ and $F$ and seek to maximize the number of copies of $H$ in an $n$-vertex graph not containing $F$ as a subgraph. We consider generalized Turán problems where the host graph is planar. In particular, we obtain the order of magnitude of the maximum number of copies of a fixed tree in a planar graph containing no even cycle of length at most $2\ell$, for all $\ell$, $\ell \geqslant 1$. We also determine the order of magnitude of the maximum number of cycles of a given length in a planar $C_4$-free graph. An exact result is given for the maximum number of $5$-cycles in a $C_4$-free planar graph. Multiple conjectures are also introduced.

Bounds on the Steiner–Wiener index of graphs

Let S be a set of vertices of a connected graph G. The Steiner distance of S is the minimum size among all connected subgraphs of G containing the vertices of S. The sum of all Steiner distances on sets of size k is called the Steiner k-Wiener index. We obtain formulae for bounds on the Steiner–Wiener index of graphs in terms of chromatic number, vertex connectivity and independence number. Also, we obtain bounds for 4-cycle free graphs.

Free Fermions Behind the Disguise

An invaluable method for probing the physics of a quantum many-body spin system is a mapping to noninteracting effective fermions. We find such mappings using only the frustration graph G of a Hamiltonian H, i.e., the network of anticommutation relations between the Pauli terms in H in a given basis. Specifically, when G is (even-hole, claw)-free, we construct an explicit free-fermion solution for H using only this structure of G, even when no Jordan–Wigner transformation exists. The solution method is generic in that it applies for any values of the couplings. This mapping generalizes both the classic Lieb–Schultz–Mattis solution of the XY model and an exact solution of a spin chain recently given by Fendley, dubbed “free fermions in disguise.” Like Fendley’s original example, the free-fermion operators that solve the model are generally highly nonlinear and nonlocal, but can nonetheless be found explicitly using a transfer operator defined in terms of the independent sets of G. The associated single-particle energies are calculated using the roots of the independence polynomial of G, which are guaranteed to be real by a result of Chudnovsky and Seymour. Furthermore, recognizing (even-hole, claw)-free graphs can be done in polynomial time, so recognizing when a spin model is solvable in this way is efficient. In a crucial step to proving our result, we additionally prove that there exists a hierarchy of commuting conserved charges for models whose frustration graphs are claw-free only, and hence these models are integrable. Finally, we give several example families of solvable and integrable models for which no Jordan–Wigner solution exists, and we give a detailed analysis of such a spin chain having 4-body couplings using this method.

Graphs with polynomially many minimal separators

We show that graphs that do not contain a theta, pyramid, prism, or turtle as an induced subgraph have polynomially many minimal separators. This result is the best possible in the sense that there are graphs with exponentially many minimal separators if only three of the four induced subgraphs are excluded. As a consequence, there is a polynomial time algorithm to solve the maximum weight independent set problem for the class of (theta, pyramid, prism, turtle)-free graphs. Since every prism, theta, and turtle contains an even hole, this also implies a polynomial time algorithm to solve the maximum weight independent set problem for the class of (pyramid, even hole)-free graphs.

Hamiltonian Cycle in K1,r-Free Split Graphs — A Dichotomy

For an optimization problem known to be NP-Hard, the dichotomy study investigates the reduction instances to determine the line separating polynomial-time solvable vs NP-Hard instances (easy vs hard instances). In this paper, we investigate the well-studied Hamiltonian cycle problem (HCYCLE), and present an interesting dichotomy result on split graphs. T. Akiyama et al. (1980) have shown that HCYCLE is NP-complete on planar bipartite graphs with maximum degree [Formula: see text]. We use this result to show that HCYCLE is NP-complete for [Formula: see text]-free split graphs. Further, we present polynomial-time algorithms for Hamiltonian cycle in [Formula: see text]-free and [Formula: see text]-free split graphs. We believe that the structural results presented in this paper can be used to show similar dichotomy result for Hamiltonian path problem and other variants of Hamiltonian cycle (path) problems.

On connected partition with degree constraints

Let s and t be nonnegative integers, and let f(s,t) be the smallest positive integer such that every f(s,t)-connected graph G admits a vertex partition (S,T) such that G[S] is s-connected and G[T] is t-connected. An (s,t)-feasible partition of graph G is a partition (S,T) of V(G) such that the minimum degree of G[S] and G[T] is at least s and t, respectively, and an (s,t)-feasible partition is said to be connected if both G[S] and G[T] are connected. Let g(s,t) be the smallest positive integer such that every graph G with minimum degree at least g(s,t) admits an (s,t)-feasible partition. Thomassen conjectured that f(s,t)=s+t+1 and g(s,t)=s+t+1. The later conjecture was confirmed by Stiebitz, while the former one is still open. In this paper, we improve Hajnal's upper bound f(s,t)≤4s+4t−13 by showing that f(s,t)≤⌈19(s−1)6⌉+⌈19(t−1)6⌉+1 if min⁡{s,t}≥3. We also show that any connected graph with an (s,t)-feasible partition also admits a connected (s,t)-feasible partition. This implies that we can extend all the results of [4,5,7–9,11,16] to connected (s,t)-feasible partitions. Finally, we show that if min⁡{s,t}≥2, then g(s,t)≤s+t−1 for {K3,K2,3,H1,H2,H3}-free graphs, and g(s,t)≤s+t for K2,3+-free graphs, where K2,3+ denote the set of graphs obtained from K2,3 by adding exactly one edge joining its two vertices, and H1, H2 and H3 are three specific graphs defined in the paper.

Structure of super strongly perfect graphs

Super strongly perfect graphs and their association with certain other classes of graphs are discussed in this paper. It is observed that every split graph is super strongly perfect. An existing result on super strongly perfect graphs is disproved providing a counter example. It is also established that if a graph G contains a cycle of odd length, then its line graph L(G) is not always super strongly perfect. Complements of cycles of length six or above are proved to be non-super strongly perfect. If a graph is strongly perfect, then it is shown that they are super strongly perfect and hence all [Formula: see text]-free graphs are super strongly perfect.

The Maximum Number of Copies of $K_{r,s}$ in Graphs Without Long Cycles or Paths

The circumference of a graph is the length of a longest cycle of it. We determine the maximum number of copies of $K_{r,s}$, the complete bipartite graph with classes sizes $r$ and $s$, in 2-connected graphs with circumference less than $k$. As corollaries of our main result, we determine the maximum number of copies of $K_{r,s}$ in $n$-vertex $P_{k}$-free and $M_k$-free graphs for all values of $n$, where $P_k$ is a path on $k$ vertices and $M_k$ is a matching on $k$ edges.

Cops and robbers on 2K2-free graphs

We prove that the cop number of any 2K2-free graph is at most 2, proving a conjecture of Sivaraman and Testa. We also show that the upper bound of 3 on the cop number of 2K1+K2-free (co-diamond–free) graphs is best possible.

On the structure and clique‐width of (4K1,C4,C6,C7)‐free graphs

AbstractWe give a complete structural description of ‐free graphs that do not contain a simplicial vertex, and we prove that such graphs have bounded clique‐width. Together with the results of Foley et al., this implies that ‐free graphs that do not contain a simplicial vertex have bounded clique‐width. Consequently, Graph Coloring can be solved in polynomial time for ‐free graphs, that is, for even‐hole‐free graphs of stability number at most three.