31 pages, 10 figures, 4 tables. The main additions to this version are the analysis of the limiting distributions of the degree of symmetry in the various cases, a new subsection 5.2 relating certain bargraphs to peakless Motzkin paths, minor corrections and improvements, and additional references
The degree of symmetry of a combinatorial object, such as a lattice path, is a measure of how symmetric the object is. It typically ranges from zero, if the object is completely asymmetric, to its size, if it is completely symmetric. We study the behavior of this statistic on Dyck paths and grand Dyck paths, with symmetry described by reflection along a vertical line through their midpoint; partitions, with symmetry given by conjugation; and certain compositions interpreted as bargraphs. We find expressions for the generating functions for these objects with respect to their degree of symmetry, and their semilength or semiperimeter, deducing in most cases that, asymptotically, the degree of symmetry has a Rayleigh or half-normal limiting distribution. The resulting generating functions are often algebraic, with the notable exception of Dyck paths, for which we conjecture that it is D-finite (but not algebraic), based on a functional equation that we obtain using bijections to walks in the plane.
A graph is path-pairable if for any pairing of its vertices there exist edge-disjoint paths joining the vertices in each pair. We investigate the behaviour of the maximum degree in path-pairable planar graphs. We show that any $n$-vertex path-pairable planar graph must contain a vertex of degree linear in $n$.
A path $v_1,v_2,\ldots,v_m$ in a graph $G$ is $degree$-$monotone$ if $deg(v_1) \leq deg(v_2) \leq \cdots \leq deg(v_m)$ where $deg(v_i)$ is the degree of $v_i$ in $G$. Longest degree-monotone paths have been studied in several recent papers. Here we consider the Ramsey type problem for degree monotone paths. Denote by $M_k(m)$ the minimum number $M$ such that for all $n \geq M$, in any $k$-edge coloring of $K_n$ there is some $1\leq j \leq k$ such that the graph formed by the edges colored $j$ has a degree-monotone path of order $m$. We prove several nontrivial upper and lower bounds for $M_k(…
A well known theorem in graph theory states that every graph $G$ on $n$ vertices and minimum degree at least $d$ contains a path of length at least $d$, and if $G$ is connected and $n\ge 2d+1$ then $G$ contains a path of length at least $2d$ (Dirac, 1952). In this article, we give an extension of Dirac's result to hypergraphs. We determine asymptotic lower bounds of the minimum degrees of 3-graphs to guarantee linear paths of specific lengths, and the lower bounds are tight up to a constant.
We give sufficient conditions under which a random graph with a specified degree sequence is symmetric or asymmetric. In the case of bounded degree sequences, our characterisation captures the phase transition of the symmetry of the random graphs. This phase transition coincides with that of the graph connectivity. However, when the maximum degree is a growing function as the number of vertices tends to infinity, our results suggest that these two thresholds do not coincide any more
A class of countable infinite graphs with unbounded vertex degree is considered. In these graphs, the vertices of large degree `repel' each other, which means that the path distance between two such vertices cannot be smaller than a certain function of their degrees. Assuming that this function increases sufficiently fast, we prove that the number of finite connected subgraphs (animals) of order N containing a given vertex x is exponentially bounded in N for N belonging to an infinite subset N_x of natural numbers. Under a less restrictive condition, the same result is obtained for the number …
A (3,4)-biregular bigraph G is a bipartite graph where all vertices in one part have degree 3 and all vertices in the other part have degree 4. A path factor of G is a spanning subgraph whose components are nontrivial paths. We prove that a simple (3,4)-biregular bigraph always has a path factor such that the endpoints of each path have degree three. Moreover we suggest a polynomial algorithm for the construction of such a path factor.
A degree monotone path in a graph $G$ is a path $P$ such that the sequence of degrees of the vertices in the order in which they appear on $P$ is monotonic. The length of the longest degree monotone path in $G$ is denoted by $mp(G)$. This parameter, inspired by the well-known Erdos-Szekeres theorem, has been studied by the authors in two earlier papers. Here we consider a saturation problem for the parameter $mp(G)$. We call $G$ saturated if, for every edge $e$ added to $G$, $mp(G+e) >mp(G)$, and we define $h(n,k)$ to be the least possible number of edges in a saturated graph $G$ on $n$ vertic…
The density of graphs with no $\ell$-path connecting equal-degree vertices: a short proof
Freie Universität Berlin（柏林自由大学）；Universität Hamburg（汉堡大学）；HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利科学院阿尔弗雷德·雷尼数学研究所）；Univ Lyon, UCBL, CNRS, INSA Lyon, LIRIS, UMR5205（里昂大学，法国国家科学研究中心，里昂国立应用科学学院）；Hamburg University of Technology（汉堡工业大学）
Let $G$ be a $k$-connected graph with $k\geq 2$. In this paper we first prove that: For two distinct vertices $x$ and $z$ in $G$, it contains a path passing through its any $k-2$ {specified} vertices with length at least the average degree of the vertices other than $x$ and $z$. Further, with this result, we prove that: If $G$ has $n$ vertices and $m$ edges, then it contains a cycle of length at least $2m/(n-1)$ passing through its any $k-1$ specified vertices. Our results generalize a theorem of Fan on the existence of long paths and a classical theorem of Erdös and Gallai on the existence of…
For a given finite graph $G$ of minimum degree at least $k$, let $G_{p}$ be a random subgraph of $G$ obtained by taking each edge independently with probability $p$. We prove that (i) if $p \ge ω/k$ for a function $ω=ω(k)$ that tends to infinity as $k$ does, then $G_p$ asymptotically almost surely contains a cycle (and thus a path) of length at least $(1-o(1))k$, and (ii) if $p \ge (1+o(1))\ln k/k$, then $G_p$ asymptotically almost surely contains a path of length at least $k$. Our theorems extend classical results on paths and cycles in the binomial random graph, obtained by taking $G$ to be …
A path $P$ in a graph $G$ is said to be a degree monotone path if the sequence of degrees of the vertices of $P$ in the order in which they appear on $P$ is monotonic. The length of the longest degree monotone path in $G$ is denoted by $mp(G)$. This parameter was first studied in an earlier paper by the authors where bounds in terms of other parameters of $G$ were obtained. In this paper we concentrate on the study of how $mp(G)$ changes under various operations on $G$. We first consider how $mp(G)$ changes when an edge is deleted, added, contracted or subdivided. We similarly consider the eff…
We shall study degree-monotone paths in graphs, a problem inspired by the celebrated theorem of Erd{ő}s-Szekeres concerning the longest monotone subsequence of a given sequence of numbers. A path P in a graph G is said to be a degree monotone path if the sequence of degrees of the vertices in P in the order they appear in P is monotonic. In this paper we shall consider these three problem related to this parameter: 1. Find bounds on $mp(G)$ in terms of other parameters of $G$. 2. Study $f(n,k)$ defined to be the maximum number of edges in a graph on $n$ vertices with $mp(G) < k$. 3. Estimate t…
Without imposing restrictions on a weighted graph's arc lengths, symmetry structures cannot be expected. But, they exist. To find them, the graphs are decomposed into a component that dictates all closed path properties (e.g., shortest and longest paths), and a superfluous component that can be removed. The simpler remaining graph exposes inherent symmetry structures that form the basis for all closed path properties. For certain asymmetric problems, the symmetry is that of three-cycles; for the general undirected setting it is a type of four-cycles; for general directed problems with asymmetr…
Leaf-to-leaf paths and cycles in degree-critical graphs
London School of Economics（伦敦政治经济学院）；University College London（伦敦大学学院）；University of Science and Technology of China（中国科学技术大学）；Tsinghua University（清华大学）；University of Oxford（牛津大学）；King’s College London（伦敦国王学院）
Let $G$ be an edge-colored graph. A heterochromatic (rainbow, or multicolored) path of $G$ is such a path in which no two edges have the same color. Let $d^c(v)$ denote the color degree and $CN(v)$ denote the color neighborhood of a vertex $v$ of $G$. In a previous paper, we showed that if $d^c(v)\geq k$ (color degree condition) for every vertex $v$ of $G$, then $G$ has a heterochromatic path of length at least $\lceil\frac{k+1}{2}\rceil$, and if $|CN(u)\cup CN(v)|\geq s$ (color neighborhood union condition) for every pair of vertices $u$ and $v$ of $G$, then $G$ has a heterochromatic path of …
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 path of a graph $G$ is called a Hamilton path if it passes through all the vertices of $G$. A graph is Hamilton-connected if any two vertices are connected by a Hamilton path. Note that any bipartite graph is not Hamilton-connected. We consider the weak version of Hamilton-connected property among bipartite graphs. A weakly Hamilton-connected graph is a balanced bipartite graph $G=(X,Y,E)$ that there is a Hamilton path between any vertex $x\in X$ and $y\in Y$. In this paper, we present some degrees, number of edges, and spectral radius conditions for a simple balanced bipartite graph to be w…
Broersma and Veldman proved that every 2-connected claw-free and $P_6$-free graph is hamiltonian. Chen et al. extended this result by proving every 2-connected claw-heavy and $P_6$-free graph is hamiltonian. On the other hand, Li et al. constructed a class of 2-connected graphs which are claw-heavy and $P_6$-\emph{o}-heavy but not hamiltonian. In this paper we further give some Ore-type degree conditions restricting to induced $P_6$'s of a 2-connected claw-heavy graph that can guarantee the graph to be hamiltonian. This improves some previous related results.
For a digraph $D$, let $δ^{0}(D) = \min \{δ^{+}(D), δ^{-}(D)\}$ be the minimum semi-degree of $D$. A set of $k$ vertex-disjoint paths, $\{P_{1}, \dots, P_{k}\}$, joining a disjoint source set $S = \{s_{1}, \dots, s_{k}\}$ and sink set $T = \{t_{1}, \dots, t_{k}\}$ is called an unpaired many-to-many $k$-disjoint directed path cover ($k$-DDPC for short) of $D$, if each $P_{j}$ joins $s_{j}$ and $t_{σ(j)}$ for some permutation $σ$ on $\{1, \dots , k\}$ and $\bigcup^{k}_{j=1} V(P_{j}) = V(D)$. In this paper, we give a new proof for the following result that every digraph $D$ with $δ^{0}(D) \geq \l…
Let $D$ be a digraph. We define the minimum semi-degree of $D$ as $δ^{0}(D) := \min \{δ^{+}(D), δ^{-}(D)\}$. Let $k$ be a positive integer, and let $S = \{s\}$ and $T = \{t_{1}, \dots ,t_{k}\}$ be any two disjoint subsets of $V(D)$. A set of $k$ internally disjoint paths joining source set $S$ and sink set $T$ that cover all vertices $D$ are called a one-to-many $k$-disjoint directed path cover ($k$-DDPC for short) of $D$. A digraph $D$ is semicomplete if for every pair $x,y$ of vertices of it, there is at least one arc between $x$ and $y$. In this paper, we prove that every semicomplete digra…
A graph $G$ is $k$-path-coverable if its vertex set $V(G)$ can be covered by $k$ or fewer vertex disjoint paths. In this paper, using the $Q$-index of a connected graph $G$, we present a tight sufficient condition for $G$ with fixed minimum degree and large order to be $k$-path-coverable.
An immersion of a graph H in another graph G is a one-to-one mapping phi:V(H)->V(G) and a collection of edge-disjoint paths in G, one for each edge of H, such that the path P_{uv} corresponding to the edge uv has endpoints phi(u) and phi(v). The immersion is strong if the paths P_{uv} are internally disjoint from phi(V(H)). We prove that every simple graph of minimum degree at least 11t+7 contains a strong immersion of the complete graph K_t. This improves on previously known bound of minimum degree at least 200t obtained by DeVos et al.
A weighted graph is a graph in which every edge is assigned a non-negative real number. In a weighted graph, the weight of a path is the sum of the weights of its edges, and the weighed degree of a vertex is the sum of the weights of the edges incident with it. In this paper we give three weighted degree conditions for the existence of heavy or Hamilton paths with one or two given end-vertices in 2-connected weighted graphs.
Motivated by questions in theoretical distributed computing, we develop the combinatorial theory of abstract simplex path subdivisions. Our main application is a short and structural proof of the theorem of Castaneda and Rajsbaum. This theorem in turn implies the solvability of the weak symmetry breaking task in the immediate snapshot wait-free model in the case when the number of processes is not a power of a prime number.
We show that for every l, there exists d_l such that every 3-edge-connected graph with minimum degree d_l can be edge-partitioned into paths of length l (provided that its number of edges is divisible by l). This improves a result asserting that 24-edge-connectivity and high minimum degree provides such a partition. This is best possible as 3-edge-connectivity cannot be replaced by 2-edge connectivity.
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 …
In 2010, Mader [W. Mader, Connectivity keeping paths in $k$-connected graphs, J. Graph Theory 65 (2010) 61-69.] proved that every $k$-connected graph $G$ with minimum degree at least $\lfloor\frac{3k}{2}\rfloor+m-1$ contains a path $P$ of order $m$ such that $G-V(P)$ is still $k$-connected. In this paper, we consider similar problem for bipartite graphs, and prove that every $k$-connected bipartite graph $G$ with minimum degree at least $k+m$ contains a path $P$ of order $m$ such that $G-V(P)$ is still $k$-connected.
For a graph $G$, $n$ denotes the order of $G$, $p$ the order of a longest path in $G$ and $c$ the order of a longest cycle. We show that if $G$ is a 2-connected graph such that $d(x)+d(y)+d(z)\ge p+2$ for all triples $x,y,z$ of independent vertices, then $c\ge p-1$. This improves results of Nash-Williams (in terms of minimum degree $δ$ and order $n$), Bondy (in terms of degree sum $σ_{3}$ and order $n$), and Enomoto, Heuvel, Kaneko and Saito (in terms of degree sum $σ_3$, order $n$ and relative length $diff(G)=p-c$).
By considering the parity of the degrees and levels of nodes in increasing trees, a new combinatorial interpretation for the coefficients of the Taylor expansions of the Jacobi elliptic functions is found. As one application of this new interpretation, a conjecture of Ma-Mansour-Wang-Yeh is solved. Unifying the concepts of increasing trees and plane trees, Lin-Ma-Ma-Zhou introduced weakly increasing trees on a multiset. A symmetry joint distribution of "even-degree nodes on odd levels" and "odd-degree nodes" on weakly increasing trees is found, extending the Schett polynomials, a generalizatio…
We confirm the following conjecture which has been proposed in [{\em Linear Algebra and its Applications}, {\bf 436} (2012), No. 5, 1425-1435.]: $$ 0.945\approx\displaystyle\lim_{n\longrightarrow \infty}σ(P_n,Z_n)=\displaystyle\lim_{n\longrightarrow \infty}σ(W_n,Z_n)=\frac{1}{2}\displaystyle\lim_{n\longrightarrow \infty}σ(P_n,W_n);\ \displaystyle\lim_{n\longrightarrow \infty}σ(C_{2n},Z_{2n})=2,$$ where $σ(G_1,G_2)=\sum_{i=1}^n |λ_i(G_1)-λ_i(G_2)|$ is the spectral distance between $n$ vertex non-isomorphic graphs $G_1$ and $G_2$ with adjacency spectra $λ_1(G_i) \geq λ_2(G_i) \geq \cdots \geq λ_…
For a graph $G$ and $p\in [0,1]$, let $G_p$ arise from $G$ by deleting every edge mutually independently with probability $1-p$. The random graph model $(K_n)_p$ is certainly the most investigated random graph model and also known as the $G(n,p)$-model. We show that several results concerning the length of the longest path/cycle naturally translate to $G_p$ if $G$ is an arbitrary graph of minimum degree at least $n-1$. For a constant $c$, we show that asymptotically almost surely the length of the longest path is at least $(1-(1+ε(c))ce^{-c})n$ for some function $ε(c)\to 0$ as $c\to \infty$, a…
Refining a classical proof of Whitney, we show that any $4$-connected planar triangulation can be decomposed into a Hamiltonian path and two trees. Therefore, every $4$-connected planar graph decomposes into three forests, one having maximum degree at most $2$. We use this result to show that any Hamiltonian planar triangulation can be decomposed into two trees and one spanning tree of maximum degree at most $3$. These decompositions improve the result of Gonçalves [Covering planar graphs with forests, one having bounded maximum degree. J. Comb. Theory, Ser. B, 100(6):729--739, 2010] that ever…
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…
Quite often real-world networks can be thought of as being symmetric, in the abstract sense that vertices can be found to have similar or equivalent structural roles. However, traditional measures of symmetry in graphs are based on their automorphism groups, which do not account for the similarity of local structures. We introduce the concept of local symmetry, which reflects the structural equivalence of the vertices' egonets. We study the emergence of asymmetry in the Erdős-Rényi random graph model and identify regimes of both asymptotic local symmetry and asymptotic local asymmetry. We find…
Let $G$ be an edge colored graph. A {\it}{rainbow path} in $G$ is a path in which all the edges are colored with distinct colors. Let $d^c(v)$ be the color degree of a vertex $v$ in $G$, i.e. the number of distinct colors present on the edges incident on the vertex $v$. Let $t$ be the maximum length of a rainbow path in $G$. Chen and Li showed that if $d^c \geq k$, for every vertex $v$ of $G$, then $t \geq \left \lceil \frac{3 k}{5}\right \rceil + 1$ (Long heterochromatic paths in edge-colored graphs, The Electronic Journal of Combinatorics 12 (2005), # R33, Pages:1-33.) Unfortunately, proof b…
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…
