In this paper, graphs under consideration are always edge-colored. We consider long heterochromatic paths in heterochromatic triangle free graphs. Two kinds of such graphs are considered, one is complete graphs with Gallai colorings, i.e., heterochromatic triangle free complete graphs; the other is heterochromatic triangle free graphs with $k$-good colorings, i.e., minimum color degree at least $k$. For the heterochromatic triangle free graphs $K_n$, we obtain that for every vertex $v\in V(K_n)$, $K_n$ has a heterochromatic $v$-path of length at least $d^c(v)$; whereas for the heterochromatic triangle free graphs $G$ we show that if, for any vertex $v\in V(G)$, $d^c(v)\geq k\geq 6$, then $G$ a heterochromatic path of length at least $\frac{3k}{4}$.
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 …
Let $G$ be an edge-colored graph. The color degree of a vertex $v$ of $G$, is defined as the number of colors of the edges incident to $v$. The color number of $G$ is defined as the number of colors of the edges in $G$. A rainbow triangle is one in which every pair of edges have distinct colors. In this paper we give some sufficient conditions for the existence of rainbow triangles in edge-colored graphs in terms of color degree, color number and edge number. As a corollary, a conjecture proposed by Li and Wang (Color degree and heterochromatic cycles in edge-colored graphs, European J. Combin…
Let $G$ be an edge-colored graph. A rainbow (heterochromatic, or multicolored) path of $G$ is such a path in which no two edges have the same color. Let the color degree of a vertex $v$ be the number of different colors that are used on the edges incident to $v$, and denote it to be $d^c(v)$. It was shown that if $d^c(v)\geq k$ for every vertex $v$ of $G$, then $G$ has a rainbow path of length at least $\min\{\lceil\frac{2k+1}{3}\rceil,k-1\}$. In the present paper, we consider the properly edge-colored complete graph $K_n$ only and improve the lower bound of the length of the longest rainbow p…
We investigate paths in Bernoulli's triangles, and derive several relations linking the partial sums of binomial coefficients to the Fibonacci numbers.
Let $G$ be a fixed graph. Two paths of length $n-1$ on $n$ vertices (Hamiltonian paths) are $G$-different if there is a subgraph isomorphic to $G$ in their union. In this paper we prove that the maximal number of pairwise triangle-different Hamiltonian paths is equal to the number of balanced bipartitions of the ground set, answering a question of Körner, Messuti and Simonyi.
Given a graph $G=(V,E)$ whose vertices have been properly coloured, we say that a path in $G$ is "colourful" if no two vertices in the path have the same colour. It is a corollary of the Gallai-Roy-Vitaver Theorem that every properly coloured graph contains a colourful path on $χ(G)$ vertices. We explore a conjecture that states that every properly coloured triangle-free graph $G$ contains an induced colourful path on $χ(G)$ vertices and prove its correctness when the girth of $G$ is at least $χ(G)$. Recent work on this conjecture by Gyárfás and Sárközy, and Scott and Seymour has shown the exi…
The suspension of the path $P_4$ consists of a $P_4$ and an additional vertex connected to each of the four vertices, and is denoted by $\hat{P_4}$. The largest number of triangles in a $\hat{P_4}$-free $n$-vertex graph is denoted by $ex(n,K_3,\hat{P_4})$. Mubayi and Mukherjee in 2020 showed that $ ex(n,K_3,\hat{P_4})= n^2/8+O(n)$. We show that for sufficiently large $n$, $ex(n,K_3,\hat{P_4})=\lfloor n^2/8\rfloor$.
A properly colored cycle (path) in an edge-colored graph is a cycle (path) with consecutive edges assigned distinct colors. A monochromatic triangle is a cycle of length $3$ with the edges assigned a same color. It is known that every edge-colored complete graph without containing monochromatic triangles always contains a properly colored Hamilton path. In this paper, we investigate the existence of properly colored cycles in edge-colored complete graphs when monochromatic triangles are forbidden. We obtain a vertex-pancyclic analogous result combined with a characterization of all the excepti…
Path pairs are a modification of parallelogram polyominoes that provide yet another combinatorial interpretation of the Catalan numbers. More generally, the number of path pairs of length $n$ and distance $δ$ corresponds to the $(n-1,δ-1)$ entry of Shapiro's so-called Catalan triangle. In this paper, we widen the notion of path pairs $(γ_1,γ_2)$ to the situation where $γ_1$ and $γ_2$ may have different lengths, and then enforce divisibility conditions on runs of vertical steps in $γ_2$. This creates a two-parameter family of integer triangles that generalize the Catalan triangle and qualify as…
Random walks are a series of up, down, and level steps that enumerate distinct paths from $(0,0)$ to $(2n,0)$, where $n$ is the semi-length of the path. We used these paths to analyze Catalan, Schröder, and Motzkin number sequences through a combination of matrix operations, quadratic functions, and inductive reasoning. Our results revealed a number of distinct patterns, some unnamed, between these number sequences and Pascal's triangle that can be explained through generating functions, first returns, group theory, and the Riordan matrix. Various proofs and properties of these number sequence…
This article introduces and investigates a refinement of alternating sign trapezoids by means of Catalan objects and Motzkin paths. Alternating sign trapezoids are a generalisation of alternating sign triangles, which were recently introduced by Ayyer, Behrend and Fischer. We show that the number of alternating sign trapezoids associated with a Catalan object (resp. Motzkin path) is a polynomial function in the length of the shorter base of the trapezoid. We also study the rational roots of the polynomials and formulate several conjectures and derive some partial results. Lastly, we deduce a c…
Connectivity keeping pendant extensions of paths in $k$-connected triangle-free graphs
Fuzhou University（福州大学）；Center for Discrete Mathematics, Fuzhou University（福州大学离散数学中心）；School of Mathematics and Statistics, Fuzhou University（福州大学数学与统计学院）；Key Laboratory for Operations Research and Cybernetics of Fujian Universities（福建省高校运筹与控制论重点实验室）
In this paper, we give a polynomial time algorithm which determines if a given triangle-free graph with no induced seven-vertex path is 3-colorable, and gives an explicit coloring if one exists.
In this paper, we give a polynomial time algorithm which determines if a given graph containing a triangle and no induced seven-vertex path is 3-colorable, and gives an explicit coloring if one exists. In previous work, we gave a polynomial time algorithm for three-coloring triangle-free graphs with no induced seven-vertex path. Combined, our work shows that three-coloring a graph with no induced seven-vertex path can be done in polynomial time.
Connectivity keeping pendant extensions of paths in $k$-connected graphs and triangle-free graphs
Center for Discrete Mathematics, Fuzhou University（福州大学离散数学中心）；Key Laboratory for Operations Research and Cybernetics of Fujian Universities（福建省高校运筹与控制论重点实验室）；School of Mathematics and Statistics, Fuzhou University（福州大学数学与统计学院）
We prove that every triangle-free graph of tree-width t has chromatic number at most ceil((t + 3)/2), and demonstrate that this bound is tight. The argument also establishes a connection between coloring graphs of tree-width t and on-line coloring of graphs of path-width t.
For any northeast path $ν$, we define two bivariate polynomials associated with the $ν$-associahedron: the $F$- and the $H$-triangle. We prove combinatorially that we can obtain one from the other by an invertible transformation of variables. These polynomials generalize the classical $F$- and $H$-triangles of F.~Chapoton in type $A$. Our proof is completely new and has the advantage of providing a combinatorial explanation of the relation between the $F$- and $H$-triangle.
Counting triangles in graphs without suspensions of paths
School of Mathematical Sciences, Anhui University（安徽大学数学科学学院）；Center for Pure Mathematics, School of Mathematical Sciences, Anhui University（安徽大学数学学院纯数学中心）
Fully Packed Loop configurations in a triangle (TFPLs) first appeared in the study of ordinary Fully Packed Loop configurations (FPLs) on the square grid where they were used to show that the number of FPLs with a given link pattern that has m nested arches is a polynomial function in m. It soon turned out that TFPLs possess a number of other nice properties. For instance, they can be seen as a generalized model of Littlewood-Richardson coefficients. We start our article by introducing oriented versions of TFPLs; their main advantage in comparison with ordinary TFPLs is that they involve only …
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…
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…
Let $G = (V, E)$ be a graph and $λ$ a non-negative integer. A graph $G$ is called a $(λ, 1)$-{\em graph} if $ (c0)$ $G$ is neither a complete graph no an edge-empty graph, $ (c1)$ every edge in $G$ belongs to exactly $λ$ triangles, and $(c2)$ every two non-adjacent vertices in $G$ are the end-vertices of exactly one two-edge path in $G$. It turns out that there are infinitely many feasible 4-tuples $(v, d, λ, 1)$ with $λ\ge 1$. On the other hand (and this is our main result), there is no $(v, d, λ, 1)$-graphs with $λ\ge 1$. As a byproduct, we obtain a generalization of the classical Friendship…
A long standing open problem in extremal graph theory is to describe all graphs that maximize the number of induced copies of a path on four vertices. The character of the problem changes in the setting of oriented graphs, and becomes more tractable. Here we resolve this problem in the setting of oriented graphs without transitive triangles.
In this paper, we prove that the class of graphs with no triangle and no induced cycle of even length at least 6 has bounded chromatic number. It is well-known that even-hole-free graphs are $χ$-bounded but we allow here the existence of $C_4$. The proof relies on the concept of Parity Changing Path, an adaptation of Trinity Changing Path which was recently introduced by Bonamy, Charbit and Thomassé to prove that graphs with no induced cycle of length divisible by three have bounded chromatic number.
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…
Given a graph $G=(V(G), E(G))$ and a set $P\subseteq V(G)$, the following concepts have been recently introduced: $(i)$ two elements of $P$ are \emph{mutually visible} if there is a shortest path between them without further elements of $P$; $(ii)$ $P$ is a \emph{mutual-visibility set} if its elements are pairwise mutually visible; $(iii)$ the \emph{mutual-visibility number} of $G$ is the size of any largest mutual-visibility set. % In this work we continue to investigate about these concepts. We first focus on mutual-visibility in Cartesian products. For this purpose, too, we introduce and in…
A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is called a rainbow path if no two edges of it are colored the same. A nontrivial connected graph $G$ is rainbow connected if for any two vertices of $G$ there is a rainbow path connecting them. The rainbow connection number of $G$, denoted by $rc(G)$, is defined as the smallest number of colors by using which there is a coloring such that $G$ is rainbow connected. In this paper, we mainly study the rainbow connection number of the line graph of a graph which contains triangles and get two sharp upper bounds for…
An orientation of a graph is semi-transitive if it is acyclic, and for any directed path $v_0\rightarrow v_1\rightarrow \cdots\rightarrow v_k$ either there is no arc between $v_0$ and $v_k$, or $v_i\rightarrow v_j$ is an arc for all $0\leq i<j\leq k$. An undirected graph is semi-transitive if it admits a semi-transitive orientation. Semi-transitive graphs generalize several important classes of graphs and they are precisely the class of word-representable graphs studied extensively in the literature. Determining if a triangle-free graph is semi-transitive is an NP-hard problem. The existence o…
The distant graph $G = G(\mathbb{P}(Z),\triangle)$ of the projective line over the ring of integers is considered. The shortest path problem in this graph is solved by use of Klein's geometric interpretation of Euclidean continued fractions. In case the minimal path is non-unique, all the possible splitting are described which allows us to give necessary and sufficient conditions for existence of a unique shortest path.
A circular Pascal array is a periodization of the familiar Pascal's triangle. Using simple operators defined on periodic sequences, we find a direct relationship between the ranges of the circular Pascal arrays and numbers of certain lattice paths within corridors, which are related to Dyck paths. This link provides new, short proofs of some nontrivial formulas found in the lattice-path literature.
Given graphs $T$ and $H$, the generalized Turán number ex$(n,T,H)$ is the maximum number of copies of $T$ in an $n$-vertex graph with no copies of $H$. Alon and Shikhelman, using a result of Erd\H os, determined the asymptotics of ex$(n,K_3,H)$ when the chromatic number of $H$ is greater than 3 and proved several results when $H$ is bipartite. We consider this problem when $H$ has chromatic number 3. Even this special case for the following relatively simple 3-chromatic graphs appears to be challenging. The suspension $\widehat H$ of a graph $H$ is the graph obtained from $H$ by adding a new v…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Iowa State University（爱荷华州立大学）；McGill University（麦吉尔大学）；Czech Technical University in Prague（布拉格捷克理工大学）
We investigate arrangements of hyperplanes whose normal vectors are given by connected subgraphs of a fixed graph. These include the resonance arrangement and certain ideal subarrangements of Weyl arrangements. We characterize those which are free, simplicial, factored, or supersolvable. In particular, such an arrangement is free if and only if the graph is a cycle, a path, an almost path, or a path with a triangle attached to it.
Fox--Grinshpun--Pach showed that every $3$-coloring of the complete graph on $n$ vertices without a rainbow triangle contains a clique of size $Ω\left(n^{1/3}\log^2 n\right)$ which uses at most two colors, and this bound is tight up to the constant factor. We show that if instead of looking for large cliques one only tries to find subgraphs of large chromatic number, one can do much better. We show that every such coloring contains a $2$-colored subgraph with chromatic number at least $n^{2/3}$, and this is best possible. We further show that for fixed positive integers $s,r$ with $s\leq r$, e…
Motivated by a recent conjecture of the first author, we prove that every properly coloured triangle-free graph of chromatic number $χ$ contains a rainbow independent set of size $\lceil\frac12χ\rceil$. This is sharp up to a factor $2$. This result and its short proof have implications for the related notion of chromatic discrepancy. Drawing inspiration from both structural and extremal graph theory, we conjecture that every triangle-free graph of chromatic number $χ$ contains an induced cycle of length $Ω(χ\logχ)$ as $χ\to\infty$. Even if one only demands an induced path of length $Ω(χ\logχ)$…
In this research, we determine the structure of (claw, bull)-free graphs. We show that every connected (claw, bull)-free graph is either an expansion of a path, an expansion of a cycle, or the complement of a triangle-free graph; where an expansion of a graph $G$ is obtained by replacing its vertices with disjoint cliques and adding all edges between cliques corresponding to adjacent vertices of $G$. This result also reveals facts about the structure of triangle-free graphs, which might be of independent interest.
In this paper we confirm a conjecture of Füredi, Jiang, and Seiver, and determine an exact formula for the Turán number $ex_3(n; P_3^3)$ of the 3-uniform linear path $P^3_3$ of length 3, valid for all $n$. It coincides with the analogous formula for the 3-uniform triangle $C^3_3$, obtained earlier by Frankl and Füredi for $n\ge 75$ and Csákány and Kahn for all $n$. In view of this coincidence, we also determine a `conditional' Turán number, defined as the maximum number of edges in a $P^3_3$-free 3-uniform hypergraph on $n$ vertices which is \emph{not} $C^3_3$-free.
For any graph $H$, let ${\rm Forb}^*(H)$ be the class of graphs with no induced subdivision of $H$. It was conjectured in [A.D. Scott, Induced trees in graphs of large chromatic number, {\em Journal of Graph Theory}, 24:297--311, 1997] that, for every graph $H$, there is a function $f_H:\mathbb{N} \rightarrow \mathbb{R}$ such that for every graph $G \in {\rm Forb}^*(H)$, $χ(G) \leq f_H(ω(G))$. We prove this conjecture for several graphs $H$, namely the paw (a triangle with a pendant edge), the bull (a triangle with two vertex-disjoint pendant edges), and what we call a "necklace," that is, a g…
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 $G$ be a graph with the usual shortest-path metric. A graph is $δ$-hyperbolic if for every geodesic triangle $T$, any side of $T$ is contained in a $δ$-neighborhood of the union of the other two sides. A graph is chordal if every induced cycle has at most three edges. In this paper we study the relation between the hyperbolicity of the graph and some chordality properties which are natural generalizations of being chordal. We find chordality properties that are weaker and stronger than being $δ$-hyperbolic. Moreover, we obtain a characterization of being hyperbolic on terms of a chordality…
High triangle density -- the graph property stating that a constant fraction of two-hop paths belong to a triangle -- is a common signature of social networks. This paper studies triangle-dense graphs from a structural perspective. We prove constructively that significant portions of a triangle-dense graph are contained in a disjoint union of dense, radius 2 subgraphs. This result quantifies the extent to which triangle-dense graphs resemble unions of cliques. We also show that our algorithm recovers planted clusterings in approximation-stable k-median instances.
An edge switch is an operation which makes a local change in a graph while maintaining the degree of every vertex. We introduce a switch move, called a triangle switch, which creates or deletes at least one triangle. Specifically, a make move is a triangle switch which chooses a path $zwvxy$ of length 4 and replaces it by a triangle $vxwv$ and an edge $yz$, while a break move performs the reverse operation. We consider various Markov chains which perform random triangle switches, and assume that every possible make or break move has positive probability of being performed. Our first result is …
Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $G$. In this paper, we consider two classes of unicyclic graphs, the star with an extra edge and the path with a triangle at one end. We provide the $2$-color Ramsey numbers for these two classes of graphs and use these to obtain general upper and lower bounds on the Gallai-Ramsey numbers.
A cycle is a graph is dominating if every edge of the graph is incident with a vertex of the cycle. In this paper, we investigate the characterization of the class of the forbidden pairs guaranteeing the existence of a dominating cycle and show the following two results: (i) Every $2$-connected $\{P_{5}, K_{4}^{-}\}$-free graph contains a longest cycle which is a dominating cycle. (ii) Every $2$-connected $\{P_{5}, W^{*}\}$-free graph contains a longest cycle which is a dominating cycle. Here $P_{5}$ is the path of order $5$, $K_{4}^{-}$ is the graph obtained from the complete graph of order $…
Let $G$ be a graph on $n$ vertices. A linear forest is a graph consisting of vertex-disjoint paths and isolated vertices. A maximum linear forest of $G$ is a subgraph of $G$ with maximum number of edges, which is a linear forest. We denote by $l(G)$ this maximum number. Let $t=\left\lfloor (k-1)/2\right \rfloor$. Recently, Ning and Wang \cite{boning} proved that if $l(G)=k-1$, then for any $k<n$ \[ e(G) \leq \max \left\{\binom{k}{2},\binom{t}{2}+t (n - t)+ c \right\}, \] where $c=0$ if $k$ is odd and $c=1$ otherwise, and the inequality is tight. In this paper, we prove that if $l(G)=k-1$ and $…
