We investigate decompositions of a graph into a small number of low diameter subgraphs. Let P(n,ε,d) be the smallest k such that every graph G=(V,E) on n vertices has an edge partition E=E_0 \cup E_1 \cup ... \cup E_k such that |E_0| \leq εn^2 and for all 1 \leq i \leq k the diameter of the subgraph spanned by E_i is at most d. Using Szemerédi's regularity lemma, Polcyn and Ruciński showed that P(n,ε,4) is bounded above by a constant depending only ε. This shows that every dense graph can be partitioned into a small number of ``small worlds'' provided that few edges can be ignored. Improving on their result, we determine P(n,ε,d) within an absolute constant factor, showing that P(n,ε,2) = Θ(n) is unbounded for ε< 1/4, P(n,ε,3) = Θ(1/ε^2) for ε> n^{-1/2} and P(n,ε,4) = Θ(1/ε) for ε> n^{-1}. We also prove that if G has large minimum degree, all the edges of G can be covered by a small number of low diameter subgraphs. Finally, we extend some of these results to hypergraphs, improving earlier work of Polcyn, Rödl, Ruciński, and Szemerédi.
Given a coloring of the edges of the complete graph on n vertices in k colors, by considering the neighbors of an arbitrary vertex it follows that there is a monochromatic diameter two subgraph on at least 1+(n-1)/k vertices. We show that for $k \ge 3$ this is asymptotically best possible, and that for k=2 there is always a monochromatic diameter two subgraph on at least $\lceil {3 \over 4} n \rceil$ vertices, which again, is best possible.
The Maximum Degree and Diameter Bounded Subgraph Problem (MaxDDBS) asks: given a host graph G, a bound on maximum degree Δ, and a diameter D, what is the largest subgraph of the host graph with degree bounded by Δand diameter bounded by D? In this paper, we investigate this problem when the host graph is the k-dimensional mesh. We provide lower bounds for the size of the largest subgraph of the mesh satisfying MaxDDBS for all k and Δ> 3 that agree with the known upper bounds up to the first two terms, and show that for Δ= 3, the lower bounds are at least the same order of growth as the upper b…
We denote by $\text{ex}(n, H, F)$ the maximum number of copies of $H$ in an $n$-vertex graph that does not contain $F$ as a subgraph. Recently, Grzesik, Győri, Salia, Tompkins considered conditions on $H$ under which $\text{ex}(n, H, K_r)$ is asymptotically attained at a blow-up of $K_{r-1}$, and proposed a conjecture. In this note we disprove their conjecture.
We give a bound on the spectral radius of subgraphs of regular graphs with given order and diameter. We give a lower bound on the smallest eigenvalue of a nonbipartite regular graph of given order and diameter.
Let the diameter cover number, $D^t_r(G)$, denote the least integer $d$ such that under any $r$-coloring of the edges of the graph $G$, there exists a collection of $t$ monochromatic subgraphs of diameter at most $d$ such that every vertex of $G$ is contained in at least one of the subgraphs. We explore the diameter cover number with two colors and two subgraphs when $G$ is a complete multipartite graph with at least three parts. We determine exactly the value of $D_2^2(G)$ for all complete tripartite graphs $G$, and almost all complete multipartite graphs with more than three parts.
Consider a graph $G$ drawn on a fixed surface, and assign to each vertex a list of colors of size at least two if $G$ is triangle-free and at least three otherwise. We prove that we can give each vertex a color from its list so that each monochromatic connected subgraph has bounded weak diameter (i.e., diameter measured in the metric of the whole graph $G$, not just the subgraph). In case that $G$ has bounded maximum degree, this implies that each connected monochromatic subgraph has bounded size. This solves a problem of Esperet and Joret for planar triangle-free graphs, and extends known res…
We wish to bring attention to a natural but slightly hidden problem, posed by Erdős and Nešetřil in the late 1980s, an edge version of the degree--diameter problem. Our main result is that, for any graph of maximum degree $Δ$ with more than $1.5 Δ^t$ edges, its line graph must have diameter larger than $t$. In the case where the graph contains no cycle of length $2t+1$, we can improve the bound on the number of edges to one that is exact for $t\in\{1,2,3,4,6\}$. In the case $Δ=3$ and $t=3$, we obtain an exact bound. Our results also have implications for the related problem of bounding the dis…
The problem of finding the largest connected subgraph of a given undirected host graph, subject to constraints on the maximum degree $Δ$ and the diameter $D$, was introduced in \cite{maxddbs}, as a generalization of the Degree-Diameter Problem. A case of special interest is when the host graph is a common parallel architecture. Here we discuss the case when the host graph is a $k$-dimensional mesh. We provide some general bounds for the order of the largest subgraph in arbitrary dimension $k$, and for the particular cases of $k=3, Δ= 4$ and $k=2, Δ= 3$, we give constructions that result in sha…
An odd (resp. even) subgraph in a multigraph is its subgraph in which every vertex has odd (resp. even) degree. We say that a multigraph can be decomposed into two odd subgraphs if its edge set can be partitioned into two sets so that both form odd subgraphs. In this paper we give a necessary and sufficient condition for the decomposability of a multigraph into two odd subgraphs. We also present a polynomial time algorithm for finding such a decomposition or showing its non-existence. We also deal with the case of the decomposability into an even subgraph and an odd subgraph.
The Ascending Subgraph Decomposition (ASD) Conjecture asserts that every graph $G$ with ${n+1\choose 2}$ edges admits an edge decomposition $G=H_1\oplus\cdots \oplus H_n$ such that $H_i$ has $i$ edges and it is isomorphic to a subgraph of $H_{i+1}$, $i=1,\ldots ,n-1$. We show that every bipartite graph $G$ with ${n+1\choose 2}$ edges such that the degree sequence $d_1,\ldots ,d_k$ of one of the stable sets satisfies $ d_{k-i}\ge n-i\; \text{for each}\; 0\le i\le k-1,$, admits an ascending subgraph decomposition with star forests. We also give a necessary condition on the degree sequence which …
We prove that every graph with maximum degree $Δ$ admits a partition of its edges into $O(\sqrtΔ)$ parts (as $Δ\to\infty$) none of which contains $C_4$ as a subgraph. This bound is sharp up to a constant factor. Our proof uses an iterated random colouring procedure.
In this paper, we consider the problem of decomposing the augmented cube $AQ_n$ into two spanning, regular, connected and pancyclic subgraphs. We prove that for $ n \geq 4$ and $ 2n - 1 = n_1 + n_2 $ with $ n_1, n_2 \geq 2,$ the augmented cube $ AQ_n$ can be decomposed into two spanning subgraphs $ H_1$ and $ H_2$ such that each $ H_i$ is $n_i$-regular and $n_i$-connected. Moreover, $H_i$ is $4$-pancyclic if $ n_i \geq 3.$
Let $k$, $λ$ and $μ$ be positive integers. A decomposition of a multigraph $ λG$ into edge-disjoint subgraphs $G_1, \ldots , G_k$ is said to be \emph{enclosed} by a decomposition of a multigraph $μH$ into edge-disjoint subgraphs $H_1, \ldots , H_k$ if $μ> λ$ and $G_i$ is a subgraph of $H_i$, $1 \leq i \leq k$. In this paper we initiate the study of when a decomposition can be enclosed by a decomposition that consists of spanning subgraphs. A decomposition of a graph is a 2-factorization if each subgraph is 2-regular and is Hamiltonian if each subgraph is a Hamiltonian cycle. Let $n$ and $m$ be…
We study a game of pursuit and evasion introduced by Seager in 2012, in which a cop searches the robber from outside the graph, using distance queries. A graph on which the cop wins is called locatable. In her original paper, Seager asked whether there exists a characterisation of the graph property of locatable graphs by either forbidden or forbidden induced subgraphs, both of which we answer in the negative. We then proceed to show that such a characterisation does exist for graphs of diameter at most 2, stating it explicitly, and note that this is not true for higher diameter. Exploring a d…
A graph $H$ is an induced subgraph of a graph $G$ if a graph isomorphic to $H$ can be obtained from $G$ by deleting vertices. Recently, there has been significant interest in understanding the unavoidable induced subgraphs for graphs of large treewidth. Motivated by this work, we consider the analogous problem for pathwidth: what are the unavoidable induced subgraphs for graphs of large pathwidth? While resolving this question in the general setting looks challenging, we prove various results for sparse graphs. In particular, we show that every graph with bounded maximum degree and sufficientl…
The conditional diameter of a connected graph $Γ=(V,E)$ is defined as follows: given a property ${\cal P}$ of a pair $(Γ_1, Γ_2)$ of subgraphs of $Γ$, the so-called \emph{conditional diameter} or ${\cal P}$-{\em diameter} measures the maximum distance among subgraphs satisfying ${\cal P}$. That is, \[ D_{\cal P}(Γ):=\max_{Γ_1, Γ_2\subset Γ} \{\partial(Γ_1, Γ_2): Γ_1, Γ_2 \quad {\rm satisfy }\quad {\cal P}\}. \] In this paper we consider the conditional diameter in which ${\cal P}$ requires that $δ(u)\ge α$ for all $ u\in V(Γ_1)$, $δ(v)\ge β$ for all $v\in V(Γ_2)$, $| V(Γ_1)| \ge s$ and $| V(Γ_…
A large class of Positional Games are defined on the complete graph on $n$ vertices. The players, Maker and Breaker, take the edges of the graph in turns, and Maker wins iff his subgraph has a given -- usually monotone -- property. Here we introduce the $d$-diameter game, which means that Maker wins iff the diameter of his subgraph is at most $d$. We investigate the biased version of the game; i.e., when the players may take more than one, and not necessarily the same number of edges, in a turn. Our main result is that we proved that the $2$-diameter game has the following surprising property:…
In this paper we present a complete characterization of the smallest sets that block all the simple spanning trees (SSTs) in a complete geometric graph. We also show that if a subgraph is a blocker for all SSTs of diameter at most 4, then it must block all simple spanning subgraphs, and in particular, all SSTs. For convex geometric graphs, we obtain an even stronger result: being a blocker for all SSTs of diameter at most 3 is already sufficient for blocking all simple spanning subgraphs.
A {\em $(d,h)$-decomposition} of a graph $G$ is an order pair $(D,H)$ such that $H$ is a subgraph of $G$ where $H$ has the maximum degree at most $h$ and $D$ is an acyclic orientation of $G-E(H)$ of maximum out-degree at most $d$. A graph $G$ is {\em $(d, h)$-decomposable} if $G$ has a $(d,h)$-decomposition. Let $G$ be a graph embeddable in a surface of nonnegative characteristic. In this paper, we prove the following results. (1) If $G$ has no chord $5$-cycles or no chord $6$-cycles or no chord $7$-cycles and no adjacent $4$-cycles, then $G$ is $(3,1)$-decomposable, which generalizes the resu…
An independent set $I_c$ is a \textit{critical independent set} if $|I_c| - |N(I_c)| \geq |J| - |N(J)|$, for any independent set $J$. The \textit{critical independence number} of a graph is the cardinality of a maximum critical independent set. This number is a lower bound for the independence number and can be computed in polynomial-time. Any graph can be decomposed into two subgraphs where the independence number of one subgraph equals its critical independence number, where the critical independence number of the other subgraph is zero, and where the sum of the independence numbers of the s…
We construct an infinite family of counterexamples to Thomassen's conjecture that the vertices of every 3-connected, cubic graph on at least 8 vertices can be colored blue and red such that the blue subgraph has maximum degree at most 1 and the red subgraph minimum degree at least 1 and contains no path on 4 vertices.
We leverage an algorithm of Deming [R.W. Deming, Independence numbers of graphs -- an extension of the Koenig-Egervary theorem, Discrete Math., 27(1979), no. 1, 23--33; MR534950] to decompose a matchable graph into subgraphs with a precise structure: they are either spanning even subdivisions of blossom pairs, spanning even subdivisions of the complete graph $K_4$, or a Kőnig-Egerváry graph. In each case, the subgraphs have perfect matchings; in the first two cases, their independence numbers are one less than their matching numbers, while the independence number of the KE subgraph equals its …
It is known that the class of all graphs not containing a graph $H$ as an induced subgraph is cop-bounded if and only if $H$ is a forest whose every component is a path. In this study, we characterize all sets $\mathscr{H}$ of graphs with some $k\in \mathbb{N}$ bounding the diameter of members of $\mathscr{H}$ from above, such that $\mathscr{H}$-free graphs, i.e. graphs with no member of $\mathscr{H}$ as an induced subgraph, are cop-bounded. This, in particular, gives a characterization of cop-bounded classes of graphs defined by a finite set of connected graphs as forbidden induced subgraphs.…
The Tree Decomposition Conjecture by Barát and Thomassen states that for every tree $T$ there exists a natural number $k(T)$ such that the following holds: If $G$ is a $k(T)$-edge-connected simple graph with size divisible by the size of $T$, then $G$ can be edge-decomposed into subgraphs isomorphic to $T$. So far this conjecture has only been verified for paths, stars, and a family of bistars. We prove a weaker version of the Tree Decomposition Conjecture, where we require the subgraphs in the decomposition to be isomorphic to graphs that can be obtained from $T$ by vertex-identifications. We…
It is known that any planar graph with diameter D has treewidth O(D), and this fact has been used as the basis for several planar graph algorithms. We investigate the extent to which similar relations hold in other graph families. We show that treewidth is bounded by a function of the diameter in a minor-closed family, if and only if some apex graph does not belong to the family. In particular, the O(D) bound above can be extended to bounded-genus graphs. As a consequence, we extend several approximation algorithms and exact subgraph isomorphism algorithms from planar graphs to other graph fam…
A theta is a graph consisting of two non-adjacent vertices and three internally disjoint paths between them, each of length at least two. For a family $\mathcal{H}$ of graphs, we say a graph $G$ is $\mathcal{H}$-free if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. We prove a conjecture of Sintiari and Trotignon, that there exists an absolute constant $c$ for which every (theta, triangle)-free graph $G$ has treewidth at most $c\log (|V(G)|)$. A construction by Sintiari and Trotignon shows that this bound is asymptotically best possible, and (theta, triangle)-free graph…
Characterizing the graph classes such that, on $n$-vertex $m$-edge graphs in the class, we can compute the diameter faster than in ${\cal O}(nm)$ time is an important research problem both in theory and in practice. We here make a new step in this direction, for some metrically defined graph classes. Specifically, a subgraph $H$ of a graph $G$ is called a retract of $G$ if it is the image of some idempotent endomorphism of $G$. Two necessary conditions for $H$ being a retract of $G$ is to have $H$ is an isometric and isochromatic subgraph of $G$. We say that $H$ is an absolute retract of some …
We study the polyhedral properties of three problems of constructing an optimal complete bipartite subgraph (a biclique) in a bipartite graph. In the first problem we consider a balanced biclique with the same number of vertices in both parts and arbitrary edge weights. In the other two problems we are dealing with unbalanced subgraphs of maximum and minimum weight with nonnegative edges. All three problems are established to be NP-hard. We study the polytopes and the cone decompositions of these problems and their 1-skeletons. We describe the adjacency criterion in 1-skeleton of the polytope …
We prove a decomposition theorem for the class of triangle-free graphs that do not contain a subdivision of the complete graph on four vertices as an induced subgraph. We prove that every graph of girth at least~5 in this class is 3-colorable.
Thomassen formulated the following conjecture: Every $3$-connected cubic graph has a red-blue vertex coloring such that the blue subgraph has maximum degree $1$ (that is, it consists of a matching and some isolated vertices) and the red subgraph has minimum degree at least $1$ and contains no $3$-edge path. We prove the conjecture for Generalized Petersen graphs. We indicate that a coloring with the same properties might exist for any subcubic graph. We confirm this statement for all subcubic trees.
A path decomposition of a graph G is a collection of edge-disjoint paths of G that covers the edge set of G. Gallai (1968) conjectured that every connected graph on n vertices admits a path decomposition of cardinality at most (n+1)/2. Seminal results towards its verification consider the graph obtained from G by removing its vertices of odd degree, which is called the E-subgraph of G. Lovász (1968) verified Gallai's Conjecture for graphs whose E-subgraphs consist of at most one vertex, and Pyber (1996) verified it for graphs whose E-subgraphs are forests. In 2005, Fan verified Gallai's Conjec…
Let $η(G)$ be the number of connected induced subgraphs in a graph $G$, and $\overline{G}$ the complement of $G$. We prove that $η(G)+η(\overline{G})$ is minimum, among all $n$-vertex graphs, if and only if $G$ has no induced path on four vertices. Since the $n$-vertex star $S_n$ with maximum degree $n-1$ is the unique tree of diameter $2$, $η(S_n)+η(\overline{S_n})$ is minimum among all $n$-vertex trees, while the maximum is shown to be achieved only by the tree whose degree sequence is $(\lceil n/2\rceil,\lfloor n/2\rfloor,1,\dots,1)$. Furthermore, we prove that every graph $G$ of order $n\g…
Tree decompositions whose trees are subgraphs: An application of Simon's factorization
LaBRI, Université de Bordeaux（波尔多大学）；Département d’Informatique, Université libre de Bruxelles（布鲁塞尔自由大学）；Department of Theoretical Computer Science, Jagiellonian University（雅盖隆大学）；Faculty of Mathematics, Natural Sciences and Information Technologies and Andrej Marušič Institute, University of Primorska（普里莫斯卡大学）；Institute of Informatics, Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw（华沙大学）
We prove that a large family of graphs which are decomposable with respect to the modular decomposition can be reconstructed from their collection of vertex-deleted subgraphs.
A decomposition of a multigraph $G$ is a partition of its edges into subgraphs $G(1), \ldots , G(k)$. It is called an $r$-factorization if every $G(i)$ is $r$-regular and spanning. If $G$ is a subgraph of $H$, a decomposition of $G$ is said to be enclosed in a decomposition of $H$ if, for every $1 \leq i \leq k$, $G(i)$ is a subgraph of $H(i)$. Feghali and Johnson gave necessary and sufficient conditions for a given decomposition of $λK_n$ to be enclosed in some $2$-edge-connected $r$-factorization of $μK_{m}$ for some range of values for the parameters $n$, $m$, $λ$, $μ$, $r$: $r=2$, $μ>λ$ an…
For $k \geq 1$ and $n \geq 2k$, the well known Kneser graph $\operatorname{KG}(n,k)$ has all $k$-element subsets of an $n$-element set as vertices; two such subsets are adjacent if they are disjoint. Schrijver constructed a vertex-critical subgraph $\operatorname{SG}(n,k)$ of $\operatorname{KG}(n,k)$ with the same chromatic number. In this paper, we compute the diameter of the graph $\operatorname{SG}(2k+r,k)$ with $r \geq 1$. We obtain an exact value of the diameter of $\operatorname{SG}(2k+r,k)$ when $r \in \{1,2\}$ or when $r \geq k-3$. For the remained cases, when $3 \leq r \leq k-4$, we o…
P. Erdős, J. Pach, R. Pollack, and Z. Tuza [J. Combin. Theory, B 47 (1989), 279--285] made conjectures for the maximum diameter of connected graphs without a complete subgraph $K_{k+1}$, which have order $n$ and minimum degree $δ$. Settling a weaker version of a problem, by strengthening the $K_{k+1}$-free condition to $k$-colorable, we solve the problem for $k=3$ and $k=4$ using a unified linear programming duality approach. The case $k=4$ is a substantial simplification of the result of É. Czabarka, P. Dankelmann, and L. A. Székely [Europ. J. Comb., 30 (2009), 1082--1089].
Let $Γ$ be a $Q$-polynomial distance-regular graph of diameter $d\geq 3$. Fix a vertex $γ$ of $Γ$ and consider the subgraph induced on the union of the last two subconstituents of $Γ$ with respect to $γ$. We prove that this subgraph is connected.
A graph is locally irregular if no two adjacent vertices have the same degree. The irregular chromatic index $χ_{\rm irr}'(G)$ of a graph $G$ is the smallest number of locally irregular subgraphs needed to edge-decompose $G$. Not all graphs have such a decomposition, but Baudon, Bensmail, Przybyło, and Woźniak conjectured that if $G$ can be decomposed into locally irregular subgraphs, then $χ_{\rm irr}'(G)\leq 3$. In support of this conjecture, Przybyło showed that $χ_{\rm irr}'(G)\leq 3$ holds whenever $G$ has minimum degree at least $10^{10}$. Here we prove that every bipartite graph $G$ whi…
We consider vertex decompositions of (di)graphs which appear in Automata Theory, and establish some their properties. Then we apply them to the problem of forbidden subgraphs.
We conjecture that finite graphs with positive Cheeger constant admit a spanning subgraph with positive Cheeger constant and girth proportional to the diameter. We prove this conjecture for regular expander graphs with large expansion. Our proof relies on the Local Lemma.
