Let G be an n-vertex graph with m edges. The degree deviation measure of G is defined as s(G)=sum v in V(G)|degG(v)-(2m/n)|, where n and m are the number of vertices and edges of G, respectively. The aim of this paper is to prove the Conjecture 4.2 of [J A de Oliveira, C S Oliveira, C Justel and N M Maia de Abreu, Measures of irregularity of graphs, Pesq. Oper. 33 (3) (2013) 383-398]. The degree deviation measure of chemical graphs under some conditions on the cyclomatic number is also computed.
Given a graph G of order n and size m, let s(G)= sum|d(u)-2m/n|, where the sum is taken over all vertices u of G. We investigate upper and lower bounds on eigenvalues of G in terms of s(G).
Let $G$ be a graph. In a famous paper Collatz and Sinogowitz had proposed to measure its deviation from regularity by the difference of the (adjacency) spectral radius and the average degree: $ε(G)=ρ(G)-\frac{2m}{n}$. We obtain here a new upper bound on $ε(G)$ which seems to consistently outperform the best known upper bound to date, due to Nikiforov. The method of proof may also be of independent interest, as we use notions from numerical analysis to re-cast the estimation of $ε(G)$ as a special case of the estimation of the difference between Rayleigh quotients of proximal vectors.
S.B. Rao conjectured in 1971 that graphic degree sequences are well quasi ordered by a relation defined in terms of the induced subgraph relation. In 2008, M. Chudnovsky and P. Seymour proved this long standing Rao's Conjecture by giving structure theorems for graphic degree sequences. In this paper, we prove and use a variant of Dickson's Lemma from commutative semigroup theory to give a short proof of the bounded degree case of Rao's Conjecture that is independent of the Chudnovsky-Seymour structure theory. In fact, we affirmatively answer two questions of N. Robertson, the first of which im…
We prove the Erd\H os--Sós conjecture for trees with bounded maximum degree and large dense host graphs. As a corollary, we obtain an upper bound on the multicolour Ramsey number of large trees whose maximum degree is bounded by a constant.
S. B. Rao conjectured that graphic sequences are well-quasi-ordered under an inclusion based on induced subgraphs. This conjecture has now been proved by Chudnovsky and Seymour. We give an independent short proof of the labelled version of the almost bounded case of S. B. Rao's conjecture, the case where we have a bound on the degree, but allow a bounded number of vertices to have unbounded degree.
In this paper, we prove: Let A be a nonnegative primitive tensor with order m and dimension n. Then its primitive degree R(A)\leq (n-1)^2+1, and the upper bound is sharp. This confirms a conjecture of Shao [7].
We conjecture that every $n$-vertex graph of minimum degree at least $\frac k2$ and maximum degree at least $2k$ contains all trees with $k$ edges as subgraphs. We prove an approximate version of this conjecture for trees of bounded degree and dense host graphs. Our work also has implications on the Erd\H os--Sós conjecture and the $\frac 23$-conjecture. We prove an approximate version of both conjectures for bounded degree trees and dense host graphs.
The energy $E(G)$ of a graph $G$ is defined as the sum of the absolute values of its eigenvalues. A graph $G$ of order $n$ is said to be hypoenergetic if $E(G)<n$. Majstorović et al. conjectured that complete bipartite graph $K_{2,3}$ is the only hypoenergetic connected quadrangle-containing graph with maximum degree $Δ\leq 3$. This paper is devoted to giving a confirmative proof to the conjecture.
A linear equation is $r$-regular, if, for every $r$-coloring of the positive integers, there exist positive integers of the same color which satisfy the equation. In 2005, Fox and Radoicic conjectured that the equation $x_1 + 2x_2 + \cdots + 2^{n-2}x_{n-1} - 2^{n-1}x_n = 0$, for any $n \geq 2$, has a degree of regularity of $n-1$, which would verify a conjecture of Rado from 1933. Rado's conjecture has since been verified with a different family of equations. In this paper, we show that Fox and Radoicic's family of equations indeed have a degree of regularity of $n-1$. We also provide a few ex…
The first degree-based entropy of a graph is the Shannon entropy of its degree sequence normalized by the degree sum. In this paper, we characterize the connected graphs with given order $n$ and size $m$ that minimize the first degree-based entropy whenever $n-1 \le m \le 2n-3,$ thus extending and proving a conjecture by Yan.
Tuza's Conjecture states that if a graph $G$ does not contain more than $k$ edge-disjoint triangles, then some set of at most $2k$ edges meets all triangles of $G$. We prove Tuza's Conjecture for all graphs $G$ having no subgraph with average degree at least $7$. As a key tool in the proof, we introduce a notion of reducible sets for Tuza's Conjecture; these are substructures which cannot occur in a minimal counterexample to Tuza's Conjecture. We also introduce weak König--Egerváry graphs, a generalization of the well-studied König--Egerváry graphs.
Erdős, Faudree, Rousseau and Schelp observed the following fact for every fixed integer $k\geq 2$: Every graph on $n\geq k-1$ vertices with at least $(k-1)(n-k+2)+{k-2\choose 2}$ edges contains a subgraph with minimum degree at least $k$. However, there are examples in which the whole graph is the only such subgraph. Erdős et al. conjectured that having just one more edge implies the existence of a subgraph on at most $(1-\varepsilon_k)n$ vertices with minimum degree at least $k$, where $\varepsilon_k>0$ depends only on $k$. We prove this conjecture, using and extending ideas of Mousset, Noeve…
In 1981, Erdős and Simonovits conjectured that for any bipartite graph $H$ we have $\mathrm{ex}(n,H)=O(n^{3/2})$ if and only if $H$ is $2$-degenerate. Later, Erdős offered 250 dollars for a proof and 500 dollars for a counterexample. In this paper, we disprove the conjecture by finding, for any $\varepsilon>0$, a $3$-regular bipartite graph $H$ with $\mathrm{ex}(n,H)=O(n^{4/3+\varepsilon})$.
Let $G$ be a simple graph with maximum degree $Δ(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>Δ(G)\lfloor \frac{1}{2}|V(H)| \rfloor$. Chetwynd and Hilton in 1986 conjectured that a graph $G$ with $Δ(G)>\frac{1}{3}|V(G)|$ has chromatic index $Δ(G)$ if and only if $G$ contains no overfull subgraph. Let $0<\varepsilon <1$ and $G$ be a large graph on $n$ vertices with minimum degree at least $\frac{1}{2}(1+\varepsilon)n$. It was shown that the conjecture holds for $G$ if $n$ is even. In this paper, the same result is proved if $n$ is odd. As far as we know, this is the first result on the con…
The Bermond-Thomassen conjecture states that, for any positive integer $r$, a digraph of minimum out-degree at least $2r-1$ contains at least $r$ vertex-disjoint directed cycles. In 2014, Bang-Jensen, Bessy and Thomass\' e proved the conjecture for tournaments. In 2010, Lichiardopol conjectured that a tournament $T$ with minimum out-degree at least $(q-1)r-1$ contains at least $r$ vertex-disjoint $q$-cycles, where integer $q\geq3$ and $r\geq1$. In this paper, we address Lichiardopol's conjecture affirmatively. In particular, the case $q=3$ implies Bermond-Thomassen conjecture for tournaments.
A simple graph $G$ is \emph{overfull} if $|E(G)|>Δ\lfloor|V(G)|/2\rfloor$. By the pigeonhole principle, every overfull graph $G$ has $χ'(G)>Δ$. The \emph{core} of a graph, denoted $G_Δ$, is the subgraph induced by its vertices of degree $Δ$. Vizing's Adjacency Lemma implies that if $χ'(G)>Δ$, then $G_Δ$ contains cycles. Hilton and Zhao conjectured that if $G_Δ$ has maximum degree 2 and $Δ\ge 4$, then $χ'(G)>Δ$ precisely when $G$ is overfull. We prove this conjecture for the case $Δ=4$.
We use an algebraic method to prove a degree version of the celebrated Erd\H os-Ko-Rado theorem: given $n>2k$, every intersecting $k$-uniform hypergraph $H$ on $n$ vertices contains a vertex that lies on at most $\binom{n-2}{k-2}$ edges. This result can be viewed as a special case of the degree version of a well-known conjecture of Erdős on hypergraph matchings. Improving the work of Bollobás, Daykin, and Erd\H os from 1976, we show that given integers $n, k, s$ with $n\ge 3k^2 s$, every $k$-uniform hypergraph $H$ on $n$ vertices with minimum vertex degree greater than $\binom{n-1}{k-1}-\binom…
S. B. Rao conjectured that graphic sequences are well-quasi-ordered under an inclusion based on induced subgraphs. This conjecture has now been settled completely by M. Chudnovsky and P. Seymour. One part of the proof proves the result for the bounded case, a result proved independently by C. J. Altomare. We give two short proofs of the bounded case of S. B. Rao's conjecture. Both the proofs use the fact that if the number of entries in an integer sequence (with even sum) is much larger than its highest term, then it is necessarily graphic.
A conjecture of Richter and Salazar about graphs that are critical for a fixed crossing number $k$ is that they have bounded bandwidth. A weaker well-known conjecture of Richter is that their maximum degree is bounded in terms of $k$. In this note we disprove these conjectures for every $k\ge 171$, by providing examples of $k$-crossing-critical graphs with arbitrarily large maximum degree.
Preferential attachment schemes, where the selection mechanism is linear and possibly time-dependent, are considered, and an infinite-dimensional large deviation principle for the sample path evolution of the empirical degree distribution is found by Dupuis-Ellis-type methods. Interestingly, the rate function, which can be evaluated, contains a term which accounts for the cost of assigning a fraction of the total degree to an "infinite" degree component, that is, when an atypical "condensation" effect occurs with respect to the degree structure. As a consequence of the large deviation results,…
An end of a graph $G$ is an equivalence class of rays, where two rays are equivalent if there are infinitely many vertex-disjoint paths between them in $G$. The degree of an end is the maximum cardinality of a collection of pairwise disjoint rays in this equivalence class. Halin conjectured that the end degree can be characterised in terms of certain typical ray configurations, which would generalise his famous \emph{grid theorem}. In particular, every end of regular uncountable degree $κ$ would contain a \emph{star of rays}, i.e.\ a configuration consisting of a central ray $R$ and $κ$ neighb…
A famous conjecture of Erdős and Sós states that every graph with average degree more than $k - 1$ contains all trees with $k$ edges as subgraphs. We prove that the Erdős-Sós conjecture holds approximately, if the size of the embedded tree is linear in the size of the graph, and the maximum degree of the tree is sublinear.
A well-known conjecture of Erdős and Sós states that every graph with average degree exceeding $m-1$ contains every tree with $m$ edges as a subgraph. We propose a variant of this conjecture, which states that every graph of maximum degree exceeding $m$ and minimum degree at least $\lfloor \frac{2m}{3}\rfloor$ contains every tree with $m$ edges. As evidence for our conjecture we show (i) for every $m$ there is a $g(m)$ such that the weakening of the conjecture obtained by replacing $m$ by $g(m)$ holds, and (ii) there is a $γ>0$ such that the weakening of the conjecture obtained by replacing $\…
Answering a question of Häggkvist and Scott, Verstraëte proved that every sufficiently large graph with average degree at least $k^2+19k+10$ contains $k$ vertex-disjoint cycles of consecutive even lengths. He further conjectured that the same holds for every graph $G$ with average degree at least $k^2+3k+2$. In this paper we prove this conjecture for $k\geq 19$ when $G$ is sufficiently large. We also show that for any $ε>0$ and large $k\geq k_ε$, average degree at least $k^2+3k-2+ε$ suffices, which is asymptotically tight for infinitely many graphs.
Let $G$ be a simple graph. A pendant path of $G$ is a path such that one of its end vertices has degree $1$, the other end has degree $\ge3$, and all the internal vertices have degree $2$. Let $p_k(G)$ be the number of pendant paths of length $k$ of $G$, and $q_k(G)$ be the number of vertices with degree $\ge3$ which are an end vertex of some pendant paths of length $k$. Motivated by the problem of characterizing dendritic trees, N. Saito and E. Woei conjectured that any graph $G$ has some Laplacian eigenvalue with multiplicity at least $p_k(G)-q_k(G)$. We prove a more general result for both …
Borodin & Kostochka conjectured that if maximum degree of a graph is greater than or equal to 9, then the chromatic number of the graph is less than or equal to maximum of ω and maximum degree minus 1. Here we prove that this Conjecture is true for {P3 UNION K1}-free graphs and {K2 UNION complement of K2}-free graphs.
Gallai's path decomposition conjecture states that the edges of any connected graph on n vertices can be decomposed into at most (n+1)/2 paths. We confirm that conjecture for all graphs with maximum degree at most five.
A famous conjecture (usually called Ryser's conjecture) that appeared in the Ph.D thesis of his student, J.~R.~Henderson [15], states that for an $r$-uniform $r$-partite hypergraph $\mathcal{H}$, the inequality $τ(\mathcal{H})\le(r-1)\cdot ν(\mathcal{H})$ always holds. This conjecture is widely open, except in the case of $r=2$, when it is equivalent to K\H onig's theorem [18], and in the case of $r=3$, which was proved by Aharoni in 2001 [3]. Here we study some special cases of Ryser's conjecture. First of all the most studied special case is when $\mathcal{H}$ is intersecting. Even for this …
Thomassen, in 1983, conjectured that for a positive integer $k$, every $2$-connected non-bipartite graph of minimum degree at least $k + 1$ contains cycles of all lengths modulo $k$. In this paper, we settle this conjecture affirmatively.
In this paper, we make progress on a question related to one of Galvin that has attracted substantial attention recently. The question is that of determining among all graphs $G$ with $n$ vertices and $Δ(G)\leq r$, which has the most complete subgraphs of size $t$, for $t\geq 3$. The conjectured extremal graph is $aK_{r+1}\cup K_b$, where $n=a(r+1)+b$ with $0\leq b\leq r$. Gan, Loh, and Sudakov proved the conjecture when $a\leq 1$, and also reduced the general conjecture to the case $t=3$. We prove the conjecture for $r\leq 6$ and also establish a weaker form of the conjecture for all $r$.
We consider large random trees under Gibbs distributions and prove a Large Deviation Principle (LDP) for the distribution of degrees of vertices of the tree. The LDP rate function is given explicitly. An immediate consequence is a Law of Large Numbers for the distribution of vertex degrees in a large random tree. Our motivation for this study comes from the analysis of RNA secondary structures.
Reed conjectured that for every graph, $χ\leq \left \lceil \frac{Δ+ ω+ 1}{2} \right \rceil$ holds, where $χ$, $ω$ and $Δ$ denote the chromatic number, clique number and maximum degree of the graph, respectively. We develop an algorithm which takes a hypothetical counterexample as input. The output discloses some hidden structures closely related to high vertex degrees. Consequently, we deduce two graph classes where Reed's Conjecture holds: One contains all graphs in which the vertices of degree at least $5$ form a stable set. The other contains all graphs in which every induced cycle of odd l…
The game of Cops and Robbers is a well known pursuit-evasion game played on graphs. It has been proved \cite{bounded_degree} that cubic graphs can have arbitrarily large cop number $c(G)$, but the known constructions show only that the set $\{c(G) \mid G \text{ cubic}\}$ is unbounded. In this paper we prove that there are arbitrarily large subcubic graphs $G$ whose cop number is at least $n^{1/2-o(1)}$ where $n=|V(G)|$. We also show that proving Meyniel's conjecture for graphs of bounded degree implies a weak Meyniel's conjecture for all graphs.
The well-known 1-2-3 Conjecture asserts that the edges of every graph without an isolated edge can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general. We prove that every graph with minimum degree $δ\geq 10^6$ can be decomposed into two subgraphs requiring just weights $1$ and $2$ for the same goal. We thus prove the so-called Standard $(2,2)$-Conjecture for graphs with sufficiently large minimum degree. The result is in particular based on applications of the Lovász Local Lemma and theorems on degree-constrained subgraphs.
For a $Δ$-regular connected graph ${\sf H}$ the problem of determining the upper tail large deviation for the number of copies of ${\sf H}$ in $\mathbb{G}(n,p)$, an Erdős-Rényi graph on $n$ vertices with edge probability $p$, has generated significant interests. For $p\ll 1$ and $np^{Δ/2} \gg (\log n)^{1/(v_{\sf H}-2)}$, where $v_{\sf H}$ is the number of vertices in ${\sf H}$, the upper tail large deviation event is believed to occur due to the presence of localized structures. In this regime the large deviation event that the number of copies of ${\sf H}$ in $\mathbb{G}(n,p)$ exceeds its exp…
In this note we establish a (non-trivial) lower bound on the degree two entry $h_2$ of a Gorenstein $h$-vector of any given socle degree $e$ and any codimension $r$. In particular, when $e=4$, that is for Gorenstein $h$-vectors of the form $h=(1,r,h_2,r,1)$, our lower bound allows us to prove a conjecture of Stanley on the order of magnitude of the minimum value, say $f(r)$, that $h_2$ may assume. In fact, we show that $$\lim_{r\to \infty} {f(r)\over r^{2/3}}= 6^{2/3}.$$ In general, we wonder whether our lower bound is sharp for all integers $e\geq 4$ and $r\geq 2$.
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 $\lfloor (n+1)/2\rfloor$. Gallai's Conjecture has been verified for many classes of graphs. In particular, Lovász (1968) verified this conjecture for graphs with at most one vertex with even degree, and Pyber (1996) verified it for graphs in which every cycle contains a vertex with odd degree. Recently, Bonamy and Perrett (2016) verified Gallai's Conjecture for g…
The deviation principles of record numbers in random walk models have not been completely investigated, especially for the non-nearest neighbor cases. In this paper, we derive the asymptotic probabilities of large and moderate deviations for the number of "weak records"(or "ladder points") in two kinds of one-dimensional non-nearest neighbor random walks. The proofs depend only on the direct analysis of random walks. We illustrate that the traditional method of analyzing the local time of Brownian motions, which is often adopted for the simple random walks, would lead to wrong conjectures for …
The upper tail problem in a random graph asks to estimate the probability that the number of copies of some fixed subgraph in an Erdős--Rényi random graph exceeds its expectation by some constant factor. There has been much exciting recent progress on this problem. We study the corresponding problem for hypergraphs, for which less is known about the large deviation rate. We present new phenomena in upper tail large deviations for sparse random hypergraphs that are not seen in random graphs. We conjecture a formula for the large deviation rate, i.e., the first order asymptotics of the log-proba…
The Jacobian Conjecture states that any locally invertible polynomial system in C^n is globally invertible with polynomial inverse. C. W. Bass et al. (1982) proved a reduction theorem stating that the conjecture is true for any degree of the polynomial system if it is true in degree three. This degree reduction is obtained with the price of increasing the dimension n. We prove here a theorem concerning partial elimination of variables, which implies a reduction of the generic case to the quadratic one. The price to pay is the introduction of a supplementary parameter 0<n'< n, parameter which r…
Let $a_1, \dots, a_n \in \mathbb{R}$ satisfy $\sum_i a_i^2 = 1$, and let $\varepsilon_1, \ldots, \varepsilon_n$ be uniformly random $\pm 1$ signs and $X = \sum_{i=1}^{n} a_i \varepsilon_i$. It is conjectured that $X = \sum_{i=1}^{n} a_i \varepsilon_i$ has $\Pr[X \geq 1] \geq 7/64$. The best lower bound so far is $1/20$, due to Oleszkiewicz. In this paper we improve this to $\Pr[X \geq 1] \geq 6/64$.
We consider polynomial maps, which we call degree $d$-linear maps, that satisfy the Jacobian condition. We prove that certain infinite families of elements, which appear in the coefficients of the formal inverse of such maps, are in the ideal determined by the Jacobian condition. Using the Cayley-Hamilton theorem, we provide expressions for these elements in terms of the generators of that ideal. We also give a combinatorial proof of the Cayley-Hamilton theorem similar to that of Straubing \cite{Straubing}. We also include results of Gröbner basis computations regarding other elements.
Large deviation behavior of the largest eigenvalue $λ_1$ of Gaussian networks (Erdős-Rényi random graphs $\mathcal{G}_{n,p}$ with i.i.d. Gaussian weights on the edges) has been the topic of considerable interest. Recently in [6,30], a powerful approach was introduced based on tilting measures by suitable spherical integrals, particularly establishing a non-universal large deviation behavior for fixed $p<1$ compared to the standard Gaussian ($p=1$) case. The case when $p\to 0$ was however completely left open with one expecting the dense behavior to hold only until the average degree is logarit…
