Odd Hole Research Articles

Graphs with girth 2ℓ+1 $2\ell +1$ and without longer odd holes are 3‐colorable

AbstractFor a number , let denote the family of graphs which have girth and have no odd hole with length greater than . Wu et al. conjectured that every graph in is 3‐colorable. Chudnovsky et al. and Wu et al., respectively, proved that every graph in and is 3‐colorable. In this paper, we prove that every graph in is 3‐colorable.

Graphs with girth 9 and without longer odd holes are 3‐colourable

AbstractFor a number , let denote the family of graphs which have girth and have no odd hole with length greater than . Wu, Xu and Xu conjectured that every graph in is 3‐colourable. Chudnovsky et al., Wu et al., and Chen showed that every graph in , and is 3‐colourable, respectively. In this paper, we prove that every graph in is 3‐colourable. This confirms Wu, Xu and Xu's conjecture.

Graphs with Girth $2\ell+1$ and Without Longer Odd Holes that Contain an Odd $K_4$-Subdivision

We say that a graph $G$ has an odd $K_4$-subdivision if some subgraph of $G$ is isomorphic to a $K_4$-subdivision and whose faces are all odd holes of $G$. For a number $\ell\geq 2$, let $\mathcal{G}_{\ell}$ denote the family of graphs which have girth $2\ell+1$ and have no odd hole with length greater than $2\ell+1$. Wu, Xu and Xu conjectured that every graph in $\bigcup_{\ell\geq2}\mathcal{G}_{\ell}$ is $3$-colorable. Recently, Chudnovsky et al. and Wu et al., respectively, proved that every graph in $\mathcal{G}_2$ and $\mathcal{G}_3$ is $3$-colorable. In this paper, we prove that no $4$-vertex-critical graph in $\bigcup_{\ell\geq5}\mathcal{G}_{\ell}$ has an odd $K_4$-subdivision. Using this result, Chen proved that all graphs in $\bigcup_{\ell\geq5}\mathcal{G}_{\ell}$ are $3$-colorable.

The quantum maxima for the basic graphs of exclusivity are not reachable in Bell scenarios.

A necessary condition for the probabilities of a set of events to exhibit Bell non-locality or Kochen-Specker contextuality is that the graph of exclusivity of the events contains induced odd cycles with five or more vertices, called odd holes, or their complements, called odd antiholes. From this perspective, events whose graph of exclusivity are odd holes or antiholes are the building blocks of contextuality. For any odd hole or antihole, any assignment of probabilities allowed by quantum theory can be achieved in specific contextuality scenarios. However, here we prove that, for any odd hole, the probabilities that attain the quantum maxima cannot be achieved in Bell scenarios. We also prove it for the simplest odd antiholes. This leads us to the conjecture that the quantum maxima for any of the building blocks cannot be achieved in Bell scenarios. This result sheds light on why the problem of whether a probability assignment is quantum is decidable, while whether a probability assignment within a given Bell scenario is quantum is, in general, undecidable. This also helps to understand why identifying principles for quantum correlations is simpler when we start by identifying principles for quantum sets of probabilities defined with no reference to specific scenarios. This article is part of the theme issue 'Quantum contextuality, causality and freedom of choice'.

Intersecting and dense restrictions of clutters in polynomial time

A clutter is a family of sets, called members, such that no member contains another. It is called intersecting if every two members intersect, but not all members have a common element. Dense clutters additionally do not have a fractional packing of value 2. We are looking at certain substructures of clutters, namely minors and restrictions. For a family of clutters we introduce a general sufficient condition such that for every clutter we can decide whether the clutter has a restriction in that set in polynomial time. It is known that the sets of intersecting and dense clutters satisfy this condition. For intersecting clutters we generalize the statement to k-wise intersecting clutters using a much simpler proof. We also give a simplified proof that a dense clutter with no proper dense minor is either a delta or the blocker of an extended odd hole. This simplification reduces the running time of the algorithm for finding a delta or the blocker of an extended odd hole minor from previously O(n4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {O}}(n^4)$$\\end{document} to O(n3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {O}}(n^3)$$\\end{document} filter oracle calls.

The Chromatic Number of a Graph with Two Odd Holes and an Odd Girth

The Chromatic Number of a Graph with Two Odd Holes and an Odd Girth

On the chromatic number of graphs of odd girth without longer odd holes

An odd hole is an induced odd cycle of length at least five. Let l≥2 be an integer, and let Gl denote the family of graphs which have girth 2l+1 and have no holes of odd length at least 2l+3. Chudnovsky and Seymour proved that every graph in G2 is three-colorable. Following the idea of Chudnovsky and Seymour, Wu, Xu and Xu proved that every graph in G3 is three-colorable. In 2022, Wu, Xu and Xu conjectured that every graph G∈⋃l≥2Gl is three-colorable. In this paper, we prove that every graph G∈Gl with radius at most l+3 is three-colorable.

Perfectly contractile graphs and quadratic toric rings

AbstractPerfect graphs form one of the distinguished classes of finite simple graphs. In 2006, Chudnovsky, Robertson, Seymour, and Thomas proved that a graph is perfect if and only if it has no odd holes and no odd antiholes as induced subgraphs, which was conjectured by Berge. We consider the class of graphs that have no odd holes, no antiholes, and no odd stretchers as induced subgraphs. In particular, every graph belonging to is perfect. Everett and Reed conjectured that a graph belongs to if and only if it is perfectly contractile. In the present paper, we discuss graphs belonging to from a viewpoint of commutative algebra. In fact, we conjecture that a perfect graph belongs to if and only if the toric ideal of the stable set polytope of is generated by quadratic binomials. Especially, we show that this conjecture is true for Meyniel graphs, perfectly orderable graphs, and clique separable graphs, which are perfectly contractile graphs.

Hole number effect on internal and discharged flow characteristics of diesel injector during transient operation

Selecting the proper number of nozzle holes is a critical matter to achieving optimal combustion in diesel engines. In recent years, short transient injections with ultra-high injection pressure have been adopted in diesel engines to achieve cleaner combustion but the effect of hole number on this transient diesel injection has not been thoroughly investigated. The current study performs an experimental observation of discharged flows from diesel injectors during transient operation using the X-ray phase-contrast imaging technique for six multi-hole injectors (three-, five-, six-, eight-, nine- and ten-hole). The observation results showed that discharged flows varied without any distinctive trends among the injectors. The largest spray dispersion angle and fastest velocity deceleration were observed in the 3-hole injector which showed an almost 3 times larger dispersion angle and 10% faster deceleration than the 10-hole injector. To understand possible factors governing such flow characteristics, three-dimensional internal flow simulations were conducted for the injectors. Unlike the results trend of discharged flows, the pressure and vorticity magnitude in the nozzle sac showed a linear change by altering the hole number. However, the hole-to-hole interaction of vortex flows varied in a complex way depending on the nozzle hole number. The nozzles with small and odd hole numbers, in general, showed a stronger hole-to-hole vortex interaction that can be the primary factor governing the discharged flow characteristics during the transient operation.

2-divisibility of Some Odd Hole Free Graphs

2-divisibility of Some Odd Hole Free Graphs

On coloring of graphs of girth $2\l+1$ without longer odd holes

A hole is an induced cycle of length at least 4. Let $\l\ge~2$ be an integer and $r=\min\{\l-1,~\lfloor{l\over~2}\rfloor+1\}$, let ${\cal~G}_l$ denote the family of graphs which have girth $2\l+1$ and have no holes of odd length at least $2\l+3$, and let $G\in~{\cal~G}_{\l}$. For a vertex $u\in~V(G)$ and a nonempty set $S\subseteq~V(G)$, let $d(u,~S)=\min\{d(u,~v):v\in~S\}$, and let $L_i(S)=\{u\in~V(G)~\mbox{~and~}d(u,~S)=i\}$ for any integer $i\ge~0$. We show that if $G[S]$ is connected and $G[L_i(S)]$ is bipartite for each $i\in\{1,~\ldots,~r\}$, then $G[L_i(S)]$ is bipartite for each $i>0$, and consequently $\chi(G)\le~4$, where $G[S]$ denotes the subgraph induced by $S$.Let $\theta^-$ be the graph obtained from the Petersen graph by deleting three vertices which induce a path, let $\theta^+$ be the graph obtained from the Petersen graph by deleting two adjacent vertices, and let $\theta$ be the graph obtained from $\theta^+$ by removing an edge incident with two vertices of degree 3. For a non-Petersen graph $G\in{\cal~G}_2$, we show that if $G$ is 3-connected and has no unstable 3-cutset then $G$ must induce either $\theta$ or $\theta^-$ but does not induce $\theta^+$. As corollaries, $\chi(G)\le~3$ for every graph $G$ of ${\cal~G}_2$ that induces neither $\theta$ nor $\theta^-$, and minimal non-3-colorable graphs of ${\cal~G}_2$ induce no $\theta^+$.

Exact Solution Algorithms for the Chordless Cycle Problem

A formulation, a heuristic, and branch-and-cut algorithms are investigated for the chordless cycle problem. This is the problem of finding a largest simple cycle for a given graph so that no edge between nonimmediately subsequent cycle vertices is contained in the graph. Leaving aside procedures based on complete enumeration, no previous exact solution algorithm appears to exist for the problem, which is relevant both in theoretical and practical terms. Extensive computational results are reported here for randomly generated graphs and for graphs originating from the literature. Under acceptable CPU times, certified optimal solutions are presented for graphs with as many as 100 vertices. Summary of Contribution: Finding chordless cycles of a graph, also known as holes, is relevant, among others, to graph theory, to the design of polyhedral based exact solution algorithms to integer programming (IP) problems, and to the practical applications that benefit from these algorithms. For instance, perfect graphs do not contain odd holes. Additionally, odd hole inequalities are valid for strengthening the formulations to numerous problems that are directly defined over graphs. Furthermore, these inequalites, in association with applicable conflict graphs, are used by all modern IP solvers to preprocess and strengthen virtually any IP formulation submitted to them.

Clean tangled clutters, simplices, and projective geometries

A clutter is clean if it has no delta or the blocker of an extended odd hole minor, and it is tangled if its covering number is two and every element appears in a minimum cover. Clean tangled clutters have been instrumental in progress towards several open problems on ideal clutters, including the τ=2 Conjecture.Let C be a clean tangled clutter. It was recently proved that C has a fractional packing of value two. Collecting the supports of all such fractional packings, we obtain what is called the core of C. The core is a duplication of the cuboid of a set of 0−1 points, called the setcore of C.In this paper, we prove three results about the setcore. First, the convex hull of the setcore is a full-dimensional polytope containing the center point of the hypercube in its interior. Secondly, this polytope is a simplex if, and only if, the setcore is the cocycle space of a projective geometry over the two-element field. Finally, if this polytope is a simplex of dimension more than three, then C has the clutter of the lines of the Fano plane as a minor.Our results expose a fascinating interplay between the combinatorics and the geometry of clean tangled clutters.

Finding a shortest even hole in polynomial time

AbstractAn even (respectively, odd) hole in a graph is an induced cycle with even (respectively, odd) length that is at least four. Bienstock proved that detecting an even (respectively, odd) hole containing a given vertex is NP‐complete. Conforti, Cornuéjols, Kapoor, and Vušković gave the first known polynomial‐time algorithm to determine whether a graph contains even holes. Chudnovsky, Kawarabayashi, and Seymour estimated that Conforti et al.'s algorithm runs in time on an ‐vertex graph and reduced the required time to . Subsequently, da Silva and Vušković, Chang and Lu, and Lai, Lu, and Thorup improved the time to , , and , respectively. The tractability of determining whether a graph contains odd holes has been open for decades until the algorithm of Chudnovsky, Scott, Seymour, and Spirkl that runs in time, which Lai et al. also reduced to . By extending Chudnovsky et al.'s techniques for detecting odd holes, Chudnovsky, Scott, and Seymour (respectively) ensured the tractability of finding a long (respectively, shortest) odd hole. They also ensured the NP‐hardness of finding a longest odd hole, whose reduction also works for finding a longest even hole. Recently, Cook and Seymour ensured the tractability of finding a long even hole. An intriguing missing piece is the tractability of finding a shortest even hole, left open for 16 years by, for example, Chudnovsky et al. and Johnson. We resolve this open problem by augmenting Chudnovsky et al.'s even‐hole detection algorithm into the first known polynomial‐time algorithm, running in time, for finding a shortest even hole in an ‐vertex graph that contains even holes.

Finding a Shortest Odd Hole

An odd hole in a graph is an induced cycle with odd length greater than 3. In an earlier paper (with Sophie Spirkl), solving a longstanding open problem, we gave a polynomial-time algorithm to test if a graph has an odd hole. We subsequently showed that, for every t , there is a polynomial-time algorithm to test whether a graph contains an odd hole of length at least t . In this article, we give an algorithm that finds a shortest odd hole, if one exists.

On chordal and perfect plane near-triangulations

A plane near-triangulation G can be decomposed into a collection of induced subgraphs, described here as the W-components of G , such that G is perfect (respectively, chordal) if and only if each of its W-components is perfect (respectively, chordal). Each W-component is a 2-connected plane near-triangulation, free of edge separators and separating triangles. Graphs satisfying these conditions will be called W-near-triangulations. A linear time decomposition of G into its W-components is achievable using known techniques from the literature. W-near-triangulations have the property that the open neighbourhood of every internal vertex induces a cycle. It follows that a W-near-triangulation H of at least five vertices is non-chordal if and only if it contains an internal vertex. This yields a local structural characterization that a plane near-triangulation G is chordal if and only if it does not contain an induced wheel of at least five vertices. For W-near-triangulations that are free of induced wheels of five vertices, we derive a similar local criterion, that depends only on the neighbourhoods of individual vertices and faces, for checking perfectness. We show that a W-near-triangulation H that is free of any induced wheel of five vertices is perfect if and only if there exists neither an internal vertex x , nor a face f such that, the neighbours of x or f induce an odd hole. The above characterization leads to a linear time algorithm for determining perfectness of this class of graphs.

Counting Weighted Independent Sets beyond the Permanent

Jerrum, Sinclair and Vigoda (2004) showed that the permanent of any square matrix can be estimated in polynomial time. This computation can be viewed as approximating the partition function of edge-weighted matchings in a bipartite graph. Equivalently, this may be viewed as approximating the partition function of vertex-weighted independent sets in the line graph of a bipartite graph. Line graphs of bipartite graphs are perfect graphs, and are known to be precisely the class of (claw, diamond, odd hole)-free graphs. So how far does the result of Jerrum, Sinclair and Vigoda extend? We first show that it extends to (claw, odd hole)-free graphs, and then show that it extends to the even larger class of (fork, odd hole)-free graphs. Our techniques are based on graph decompositions, which have been the focus of much recent work in structural graph theory, and on structural results of Chvatal and Sbihi (1988), Maffray and Reed (1999) and Lozin and Milanic (2008).

A local characterization for perfect plane near-triangulations

We derive a local criterion for a plane near-triangulated graph to be perfect. It is shown that a plane near-triangulated graph is perfect if and only if it does not contain either a vertex, an edge or a triangle, the neighbourhood of which has an odd hole as its boundary. The characterization leads to an O(n2) algorithm for checking perfectness of plane near-triangulations.

Intersecting Restrictions in Clutters

A clutter is intersecting if the members do not have a common element yet every two members intersect. It has been conjectured that for clutters without an intersecting minor, total primal integrality and total dual integrality of the corresponding set covering linear system must be equivalent. In this paper, we provide a polynomial characterization of clutters without an intersecting minor. One important class of intersecting clutters comes from projective planes, namely the deltas, while another comes from graphs, namely the blockers of extended odd holes. Using similar techniques, we provide a polynomial algorithm for finding a delta or the blocker of an extended odd hole minor in a given clutter. This result is quite surprising as the same problem is NP-hard if the input were the blocker instead of the clutter.

Induced subgraphs of graphs with large chromatic number. VII. Gyárfás' complementation conjecture

Abstract A class of graphs is χ-bounded if there is a function f such that χ ( G ) ≤ f ( ω ( G ) ) for every induced subgraph G of every graph in the class, where χ , ω denote the chromatic number and clique number of G respectively. In 1987, Gyarfas conjectured that for every c, if C is a class of graphs such that χ ( G ) ≤ ω ( G ) + c for every induced subgraph G of every graph in the class, then the class of complements of members of C is χ-bounded. We prove this conjecture. Indeed, more generally, a class of graphs is χ -bounded if it has the property that no graph in the class has c + 1 odd holes, pairwise disjoint and with no edges between them. The main tool is a lemma that if C is a shortest odd hole in a graph, and X is the set of vertices with at least five neighbours in V ( C ) , then there is a three-vertex set that dominates X.