Free Bipartite Graphs Research Articles

On Girth and the Parameterized Complexity of Token Sliding and Token Jumping

In the Token Jumping problem we are given a graph $$G = (V,E)$$ and two independent sets S and T of G, each of size $$k \ge 1$$ . The goal is to determine whether there exists a sequence of k-sized independent sets in G, $$\langle S_0, S_1, \ldots , S_\ell \rangle$$ , such that for every i, $$|S_i| = k$$ , $$S_i$$ is an independent set, $$S = S_0$$ , $$S_\ell = T$$ , and $$|S_i \varDelta S_{i+1}| = 2$$ . In other words, if we view each independent set as a collection of tokens placed on a subset of the vertices of G, then the problem asks for a sequence of independent sets which transforms S to T by individual token jumps which maintain the independence of the sets. This problem is known to be PSPACE-complete on very restricted graph classes, e.g., planar bounded degree graphs and graphs of bounded bandwidth. A closely related problem is the Token Sliding problem, where instead of allowing a token to jump to any vertex of the graph we instead require that a token slides along an edge of the graph. Token Sliding is also known to be PSPACE-complete on the aforementioned graph classes. We investigate the parameterized complexity of both problems on several graph classes, focusing on the effect of excluding certain cycles from the input graph. In particular, we show that both Token Sliding and Token Jumping are fixed-parameter tractable on $$C_4$$ -free bipartite graphs when parameterized by k. For Token Jumping, we in fact show that the problem admits a polynomial kernel on $$\{C_3,C_4\}$$ -free graphs. In the case of Token Sliding, we also show that the problem admits a polynomial kernel on bipartite graphs of bounded degree. We believe both of these results to be of independent interest. We complement these positive results by showing that, for any constant $$p \ge 4$$ , both problems are W[1]-hard on $$\{C_4, \dots , C_p\}$$ -free graphs and Token Sliding remains W[1]-hard even on bipartite graphs.

The strong clique index of a graph with forbidden cycles

AbstractGiven a graph , the strong clique index of , denoted , is the maximum size of a set of edges such that every pair of edges in has distance at most 2 in the line graph of . As a relaxation of the renowned Erdős–Nešetřil conjecture regarding the strong chromatic index, Faudree et al. suggested investigating the strong clique index, and conjectured a quadratic upper bound in terms of the maximum degree. Recently, Cames van Batenburg, Kang, and Pirot conjectured a linear upper bound in terms of the maximum degree for graphs without even cycles. Namely, if is a ‐free graph with , then , and if is a ‐free bipartite graph, then . We prove the second conjecture in a stronger form, by showing that forbidding all odd cycles is not necessary. To be precise, we show that a ‐free graph with satisfies , when either or and is also ‐free. Regarding the first conjecture, we prove an upper bound that is off by the constant term. Namely, for , we prove that a ‐free graph with satisfies . This improves some results of Cames van Batenburg, Kang, and Pirot.

Exact bipartite Turán numbers of large even cycles

AbstractThe bipartite Turán number, denoted by , is the maximum number of edges in an ‐free bipartite graph with two parts of sizes and , respectively. In this article, we prove for any positive integers with , which confirms (in a strong form) the unsolved case of a conjecture of Győri. Let be a bipartite graph with and . Suppose that for some . We prove that contains cycles of all even lengths from 4 to if . This result sharpens a theorem on long cycles in bipartite graphs due to Jackson. As a main tool, for a longest cycle in a bipartite graph, we prove an upper bound of the number of edges which are incident with at most one vertex in , which is a bipartite analogue of a classical theorem of Bondy. Our proof technique is structural.

On the Extremal Number of Subdivisions

Abstract One of the cornerstones of extremal graph theory is a result of Füredi, later reproved and given due prominence by Alon, Krivelevich, and Sudakov, saying that if $H$ is a bipartite graph with maximum degree $r$ on one side, then there is a constant $C$ such that every graph with $n$ vertices and $C n^{2 - 1/r}$ edges contains a copy of $H$. This result is tight up to the constant when $H$ contains a copy of $K_{r,s}$ with $s$ sufficiently large in terms of $r$. We conjecture that this is essentially the only situation in which Füredi’s result can be tight and prove this conjecture for $r = 2$. More precisely, we show that if $H$ is a $C_4$-free bipartite graph with maximum degree $2$ on one side, then there are positive constants $C$ and $\delta $ such that every graph with $n$ vertices and $C n^{3/2 - \delta }$ edges contains a copy of $H$. This answers a question of Erd̋s from 1988. The proof relies on a novel variant of the dependent random choice technique which may be of independent interest.

Prime graphs, matchings and the Castelnuovo-Mumford regularity

We demonstrate the effectiveness of prime graphs for the calculation of the (Castelnuovo-Mumford) regularity of graphs. Such a notion allows us to reformulate the regularity as a generalized induced matching problem and perform regularity calculations in specific graph classes, including $(C_3,P_5)$-free graphs, $P_6$-free bipartite graphs and all Cohen-Macaulay graphs of girth at least five. In particular, we verify that the five cycle graph $C_5$ is the unique connected graph satisfying the inequality $im (G)\lt \mbox {reg}(G)=m (G)$. In addition, we prove that, for each integer $n\geq 1$, there exists a vertex decomposable perfect prime graph $G_n$ with $\mbox {reg}(G_n)=n$.

On Forbidden Induced Subgraphs for Unit Disk Graphs

A unit disk graph is the intersection graph of disks of equal radii in the plane. The class of unit disk graphs is hereditary, and therefore admits a characterization in terms of minimal forbidden induced subgraphs. In spite of quite active study of unit disk graphs very little is known about minimal forbidden induced subgraphs for this class. We found only finitely many minimal non-unit disk graphs in the literature. In this paper we study in a systematic way forbidden induced subgraphs for the class of unit disk graphs. We develop several structural and geometrical tools, and use them to reveal infinitely many new minimal non-unit disk graphs. Further we use these results to investigate structure of co-bipartite unit disk graphs. In particular, we give structural characterization of those co-bipartite unit disk graphs whose edges between parts form a \(C_4\)-free bipartite graph, and show that bipartite complements of these graphs are also unit disk graphs. Our results lead us to propose a conjecture that the class of co-bipartite unit disk graphs is closed under bipartite complementation.

List H-Coloring a Graph by Removing Few Vertices

In the deletion version of the list homomorphism problem, we are given graphs G and H, a list $$L(v)\subseteq V(H)$$ for each vertex $$v\in V(G)$$ , and an integer k. The task is to decide whether there exists a set $$W \subseteq V(G)$$ of size at most k such that there is a homomorphism from $$G {\setminus } W$$ to H respecting the lists. We show that DL-Hom( $${H}$$ ), parameterized by k and |H|, is fixed-parameter tractable for any $$(P_6,C_6)$$ -free bipartite graph H; already for this restricted class of graphs, the problem generalizes Vertex Cover, Odd Cycle Transversal, and Vertex Multiway Cut parameterized by the size of the cutset and the number of terminals. We conjecture that DL-Hom( $${H}$$ ) is fixed-parameter tractable for the class of graphs H for which the list homomorphism problem (without deletions) is polynomial-time solvable; by a result of Feder et al. (Combinatorica 19(4):487–505, 1999), a graph H belongs to this class precisely if it is a bipartite graph whose complement is a circular arc graph. We show that this conjecture is equivalent to the fixed-parameter tractability of a single fairly natural satisfiability problem, Clause Deletion Chain-SAT.

Bipartite rigidity

We develop a bipartite rigidity theory for bipartite graphs parallel to the classical rigidity theory for general graphs, and define for two positive integers k , l k,l the notions of ( k , l ) (k,l) -rigid and ( k , l ) (k,l) -stress free bipartite graphs. This theory coincides with the study of Babson–Novik’s balanced shifting restricted to graphs. We establish bipartite analogs of the cone, contraction, deletion, and gluing lemmas, and apply these results to derive a bipartite analog of the rigidity criterion for planar graphs. Our result asserts that for a planar bipartite graph G G its balanced shifting, G b G^b , does not contain K 3 , 3 K_{3,3} ; equivalently, planar bipartite graphs are generically ( 2 , 2 ) (2,2) -stress free. We also discuss potential applications of this theory to Jockusch’s cubical lower bound conjecture and to upper bound conjectures for embedded simplicial complexes.

Weighted coloring on planar, bipartite and split graphs: Complexity and approximation

We study complexity and approximation of min weighted node coloring in planar, bipartite and split graphs. We show that this problem is NP-hard in planar graphs, even if they are triangle-free and their maximum degree is bounded above by 4. Then, we prove that min weighted node coloring is NP-hard in P 8 -free bipartite graphs, but polynomial for P 5 -free bipartite graphs. We next focus on approximability in general bipartite graphs and improve earlier approximation results by giving approximation ratios matching inapproximability bounds. We next deal with min weighted edge coloring in bipartite graphs. We show that this problem remains strongly NP-hard, even in the case where the input graph is both cubic and planar. Furthermore, we provide an inapproximability bound of 7 / 6 − ε , for any ε > 0 and we give an approximation algorithm with the same ratio. Finally, we show that min weighted node coloring in split graphs can be solved by a polynomial time approximation scheme.

A New Algorithm for On-line Coloring Bipartite Graphs

We first show that for any bipartite graph H with at most five vertices there exists an on-line competitive algorithm for the class of H-free bipartite graphs. We then analyze the performance of an on-line algorithm for coloring bipartite graphs on various subfamilies. The algorithm yields new upper bounds for the on-line chromatic number of bipartite graphs. We prove that the algorithm is on-line competitive for $P_7$-free bipartite graphs, i.e., that do not contain an induced path on seven vertices. The number of colors used by the on-line algorithm for $P_6$-free and $P_7$-free bipartite graphs is, respectively, bounded by roughly twice and roughly eight times the on-line chromatic number. In contrast, it is known that there exists no competitive on-line algorithm to color $P_6$-free (or $P_7$-free) bipartite graphs, i.e., for which the number of colors is bounded by any function depending only on the chromatic number.

Counterexample to a conjecture of Györi on [formula omitted]-free bipartite graphs

A counterexample on a conjecture of Györi related with C 2 l -free bipartite graphs is described.

Linear time recognition of Weak bisplit graphs

Abstract We introduced in [4]a new decomposition scheme for bipartite graphs that we called canonical decomposition. Weak bisplit graphs are totally decomposable following this decomposition We give here the frame work for obtaining linear recognition algorithms for Weak bisplit graphs as well as for two sub families of this class the P-6 free bipartite graphs and the bicographs We conclude by presenting a list of NP-complete optimization problems for general bipartite graphs which they have efficient solutions when dealing with weak bisplit graphs

On canonical decomposition of bipartite graphs

Abstract We introduced in [5] a new decomposition scheme for bipartite graphs that we called canonical decomposition. Weak-bisplit graphs are totally decomposable following this decomposition. We give here the framework for obtaining linear recognition algorithms for Weak-bisplit graphs as well as for two subfamilies of this class, the P 6 -free bipartite graphs and the bi-cographs. We conclude by presenting a list of NP-complete optimization problems for general bipartite graphs which they have efficient solutions when dealing with weak-bisplit graphs.

A note on the interval number of a graph

Three results on the interval number of a graph on n vertices are presented. (1) The interval number of almost every graph is between n 4lg n and n 4 (this also holds for almost every bipartite graph). (2) There exist K m, m -free bipartite graphs with interval number at least c( m) n 1−2/( m+1) /lg n, which can be improved to n /4+o( n ) for m=2 and (n/2) 2 3 / lg n for m=3. (3) There exists a regular graph of girth at least g with interval number at least 1 2 ((n−1)/2) 1/(g−2) .