The Erdős--Gallai criteria for recognizing degree sequences of simple graphs involve a system of inequalities. Given a fixed degree sequence, we consider the list of differences of the two sides of these inequalities. These differences have appeared in varying contexts, including characterizations of the split and threshold graphs, and we survey their uses here. Then, enlarging upon properties of these graph families, we show that both the last term and the maximum term of the principal Erdős--Gallai differences of a degree sequence are preserved under graph complementation and are monotonic under the majorization order and Rao's order on degree sequences.
Combining Pósa's rotation lemma with a technique of Kopylov in a novel approach, we prove a generalization of the Erdős-Gallai theorems on cycles and paths. This implies a clique version of the Erdős-Gallai stability theorems and also provides alternative proofs for some recent results.
A keyring is a graph obtained from a cycle by appending $r\ge0$ leaves to one of its vertices. Sidorenko proved an Erdős-Gallai-type theorem: Every graph of order $n$ and size more than $\frac{(k-1)n}{2}$ contains a keyring of size at least $k$ and with $r$ leaves for $r\le\frac{k-1}2$ (Theorem 1.4, An Erdős-Gallai-type theorem for keyrings, Graphs Combin., 2018). In this note, we show that Sidorenko's theorem holds for larger $r$ and so complete the Erdős-Gallai-type theorem for keyrings.
A classical result of Erdős and Gallai determines the maximum size $m(n,ν)$ of a graph $G$ of order $n$ and matching number $νn$. We show that $G$ has factorially many maximum matchings provided that its size is sufficiently close to $m(n,ν)$.
While investigating odd-cycle free hypergraphs, Győri and Lemons introduced a colored version of the classical theorem of Erdős and Gallai on $P_k$-free graphs. They proved that any graph $G$ with a proper vertex coloring and no path of length $2k+1$ with endpoints of different colors has at most $2kn$ edges. We show that Erdős and Gallai's original sharp upper bound of $kn$ holds for their problem as well. We also introduce a version of this problem for trees and present a generalization of the Erdős-Sós conjecture.
Recently, variants of many classical extremal theorems have been proved in the random environment. We, complementing existing results, extend the Erdős-Gallai Theorem in random graphs. In particular, we determine, up to a constant factor, the maximum number of edges in a $P_n$-free subgraph of $G(N,p)$, practically for all values of $N,n$ and $p$. Our work is also motivated by the recent progress on the size-Ramsey number of paths.
A keyring is a graph obtained by appending $r \geq 1$ leaves to one of the vertices of a cycle. We prove that for every $r \leq (k-1)/2$, any graph with average degree more than $k-1$ contains a keyring with $r$ leaves and at least $k$ edges.
A path in an edge-colored graph is called \emph{a conflict-free path} if there exists a color used on only one of its edges. An edge-colored graph is called \emph{conflict-free connected} if there is a conflict-free path between each pair of distinct vertices. The \emph{conflict-free connection number} of a connected graph $G$, denoted by $\mathit{cfc}(G)$, is defined as the smallest number of colors that are required to make $G$ conflict-free connected. In this paper, we obtain Erdös-Gallai-type results for the conflict-free connection numbers of graphs.
Erdős, Gallai, and Tuza posed the following problem: given an $n$-vertex graph $G$, let $τ_1(G)$ denote the smallest size of a set of edges whose deletion makes $G$ triangle-free, and let $α_1(G)$ denote the largest size of a set of edges containing at most one edge from each triangle of $G$. Is it always the case that $α_1(G) + τ_1(G) \leq n^2/4$? We have two main results. We first obtain the upper bound $α_1(G) + τ_1(G) \leq 5n^2/16$, as a partial result towards the Erdős--Gallai--Tuza conjecture. We also show that always $α_1(G) \leq n^2/2 - m$, where $m$ is the number of edges in $G$; this…
We give characterizations of the structure and degree sequences of hereditary unigraphs, those graphs for which every induced subgraph is the unique realization of its degree sequence. The class of hereditary unigraphs properly contains the threshold and matrogenic graphs, and the characterizations presented here naturally generalize those known for these other classes of graphs. The degree sequence characterization of hereditary unigraphs makes use of the list of values $k$ for which the $k$th Erdős--Gallai inequality holds with equality for a graphic sequence. Using the canonical decompositi…
A graph is said to be {\it total-colored} if all the edges and the vertices of the graph are colored. A total-coloring of a graph is a {\it total monochromatically-connecting coloring} ({\it TMC-coloring}, for short) if any two vertices of the graph are connected by a path whose edges and internal vertices have the same color. For a connected graph $G$, the {\it total monochromatic connection number}, denoted by $tmc(G)$, is defined as the maximum number of colors used in a TMC-coloring of $G$. In this paper, we study two kinds of Erdős-Gallai-type problems for $tmc(G)$ and completely solve th…
A path in an edge-colored graph is called a \emph{monochromatic path} if all the edges on the path are colored the same. An edge-coloring of $G$ is a \emph{monochromatic connection coloring} (MC-coloring, for short) if there is a monochromatic path joining any two vertices in $G$. The \emph{monochromatic connection number}, denoted by $mc(G)$, is defined to be the maximum number of colors used in an MC-coloring of a graph $G$. These concepts were introduced by Caro and Yuster, and they got some nice results. In this paper, we will study two kinds of Erdős-Gallai-type problems for $mc(G)$, and …
The Erdős-Gallai Theorem states that every graph of average degree more than $l-2$ contains a path of order $l$ for $l\ge 2$. In this paper, we obtain a stability version of the Erdős-Gallai Theorem in terms of minimum degree. Let $G$ be a connected graph of order $n$ and $F=(\bigcup_{i=1}^kP_{2a_i})\bigcup(\bigcup_{i=1}^lP_{2b_i+1})$ be $k+l$ disjoint paths of order $2a_1, \ldots, 2a_{k}, 2b_1+1, \ldots, 2b_l+1,$ respectively, where $k\ge 0$, $0\le l\le 2$, and $k+l\geq 2$. If the minimum degree $δ(G)\ge \sum_{i=1}^ka_i+\sum_{i=1}^lb_i-1$, then $F\subseteq G$ except several classes of graphs …
Let $G$ be a nontrivial connected and edge-colored graph. An edge-cut $R$ of $G$ is called a rainbow cut if no two edges of it are colored with a same color. An edge-colored graph $G$ is called rainbow disconnected if for every two distinct vertices $u$ and $v$ of $G$, there exists a $u-v$ rainbow cut separating them. For a connected graph $G$, the rainbow disconnection number of $G$, denoted by $rd(G)$, is defined as the smallest number of colors that are needed in order to make $G$ rainbow disconnected. In this paper, we will study the Erdős-Gallai-type results for $rd(G)$, and completely so…
A well-known theorem of Erdős and Gallai asserts that a graph with no path of length $k$ contains at most $\frac{1}{2}(k-1)n$ edges. Recently Győri, Katona and Lemons gave an extension of this result to hypergraphs by determining the maximum number of hyperedges in an $r$-uniform hypergraph containing no Berge path of length $k$ for all values of $r$ and $k$ except for $k=r+1$. We settle the remaining case by proving that an $r$-uniform hypergraph with more than $n$ hyperedges must contain a Berge path of length $r+1$.
For an edge-colored graph $G$, we call an edge-cut $M$ of $G$ monochromatic if the edges of $M$ are colored with a same color. The graph $G$ is called monochromatically disconnected if any two distinct vertices of $G$ are separated by a monochromatic edge-cut. The monochromatic disconnection number, denoted by $md(G)$, of a connected graph $G$ is the maximum number of colors that are allowed to make $G$ monochromatically disconnected. In this paper, we solve the Erdős-Gallai-type problems for the monochromatic disconnection, and give the monochromatic disconnection numbers for four graph produ…
The Erdős--Gallai Theorem states that for $k \geq 3$, any $n$-vertex graph with no cycle of length at least $k$ has at most $\frac{1}{2}(k-1)(n-1)$ edges. A stronger version of the Erdős--Gallai Theorem was given by Kopylov: If $G$ is a 2-connected $n$-vertex graph with no cycle of length at least $k$, then $e(G) \leq \max\{h(n,k,2),h(n,k,\lfloor \frac{k-1}{2}\rfloor)\}$, where $h(n,k,a) := {k - a \choose 2} + a(n - k + a)$. Furthermore, Kopylov presented the two possible extremal graphs, one with $h(n,k,2)$ edges and one with $h(n,k,\lfloor \frac{k-1}{2}\rfloor)$ edges. In this paper, we comp…
For a 2-connected graph $G$ on $n$ vertices and two vertices $x,y\in V(G)$, we prove that there is an $(x,y)$-path of length at least $k$ if there are at least $\frac{n-1}{2}$ vertices in $V(G)\backslash \{x,y\}$ of degree at least $k$. This strengthens a well-known theorem due to Erdős and Gallai in 1959. As the first application of this result, we show that a 2-connected graph with $n$ vertices contains a cycle of length at least $2k$ if it has at least $\frac{n}{2}+k$ vertices of degree at least $k$. This confirms a 1975 conjecture made by Woodall. As another applications, we obtain some re…
The famous Erdős-Gallai Theorem on the Turán number of paths states that every graph with $n$ vertices and $m$ edges contains a path with at least $\frac{2m}{n}$ edges. In this note, we first establish a simple but novel extension of the Erdős-Gallai Theorem by proving that every graph $G$ contains a path with at least $\frac{(s+1)N_{s+1}(G)}{N_{s}(G)}+s-1$ edges, where $N_j(G)$ denotes the number of $j$-cliques in $G$ for $1\leq j\leqω(G)$. We also construct a family of graphs which shows our extension improves the estimate given by Erdős-Gallai Theorem. Among applications, we show, for examp…
The Erdős-Gallai Theorem states that for $k \geq 2$, every graph of average degree more than $k - 2$ contains a $k$-vertex path. This result is a consequence of a stronger result of Kopylov: if $k$ is odd, $k=2t+1\geq 5$, $n \geq (5t-3)/2$, and $G$ is an $n$-vertex $2$-connected graph with at least $h(n,k,t) := {k-t \choose 2} + t(n -k+ t)$ edges, then $G$ contains a cycle of length at least $k$ unless $G = H_{n,k,t} := K_n - E(K_{n - t})$. In this paper we prove a stability version of the Erdős-Gallai Theorem: we show that for all $n \geq 3t > 3$, and $k \in \{2t+1,2t + 2\}$, every $n$-vertex…
For fixed $p$ and $q$, an edge-coloring of the complete graph $K_n$ is said to be a $(p, q)$-coloring if every $K_p$ receives at least $q$ distinct colors. The function $f(n, p, q)$ is the minimum number of colors needed for $K_n$ to have a $(p, q)$-coloring. This function was introduced about 45 years ago, but was studied systematically by Erdős and Gyárfás in 1997, and is now known as the Erdős-Gyárfás function. In this paper, we study $f(n, p, q)$ with respect to Gallai-colorings, where a Gallai-coloring is an edge-coloring of $K_n$ without rainbow triangles. Combining the two concepts, we …
Erdős, Gallai, and Tuza posed the following problem: given an $n$-vertex graph $G$, let $τ_1(G)$ denote the smallest size of a set of edges whose deletion makes $G$ triangle-free, and let $α_1(G)$ denote the largest size of a set of edges containing at most one edge from each triangle of $G$. Is it always the case that $α_1(G) + τ_1(G) \leq n^2/4$? We also consider a variant on this conjecture: if $τ_B(G)$ is the smallest size of an edge set whose deletion makes $G$ bipartite, does the stronger inequality $α_1(G) + τ_B(G) \leq n^2/4$ always hold? By considering the structure of a minimal count…
In 1959, Erdős and Gallai proved that every graph G with average vertex degree ad(G)\geq 2 contains a cycle of length at least ad(G). We provide an algorithm that for k\geq 0 in time 2^{O(k)} n^{O(1)} decides whether a 2-connected n-vertex graph G contains a cycle of length at least ad(G)+k. This resolves an open problem explicitly mentioned in several papers. The main ingredients of our algorithm are new graph-theoretical results interesting on their own.
Extensions of Erdős-Gallai Theorem for general hypergraphs are well studied. In this work, we prove the extension of Erdős-Gallai Theorem for linear hypergraphs. In particular, we show that the number of hyperedges in an $n$-vertex $3$-uniform linear hypergraph, without a Berge path of length $k$ as a subgraph is at most $\frac{(k-1)}{6}n$ for $k\geq 4$.
In the 1960s, Erdős and Gallai conjectured that the edge set of every graph on n vertices can be partitioned into O(n) cycles and edges. They observed that one can easily get an O(n log n) upper bound by repeatedly removing the edges of the longest cycle. We make the first progress on this problem, showing that O(n log log n) cycles and edges suffice. We also prove the Erdős-Gallai conjecture for random graphs and for graphs with linear minimum degree.
An $r$-quasiplanar graph is a graph drawn in the plane with no $r$ pairwise crossing edges. Let $s \geq 3$ be an integer and $r=2^s$. We prove that there is a constant $C$ such that every $r$-quasiplanar graph with $n \geq r$ vertices has at most $n\left(Cs^{-1}\log n\right)^{2s-4}$ edges. A graph whose vertices are continuous curves in the plane, two being connected by an edge if and only if they intersect, is called a string graph. We show that for every $ε>0$, there exists $δ>0$ such that every string graph with $n$ vertices, whose chromatic number is at least $n^ε$ contains a clique of siz…
A Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai $k$-coloring is a Gallai coloring that uses $k$ colors. We study Ramsey-type problems in Gallai colorings. Given an integer $k\ge1$ and a graph $H$, the Gallai-Ramsey number $GR_k(H)$ is the least positive integer $n$ such that every Gallai $k$-coloring of the complete graph on $n$ vertices contains a monochromatic copy of $H$. It turns out that $GR_k(H)$ is more well-behaved than the classical Ramsey number $R_k(H)$. However, finding exact values of $GR_k (H)$ is far from trivial. In this …
We study Ramsey-type problems in Gallai-colorings. Given a graph $G$ and an integer $k\ge1$, the Gallai-Ramsey number $gr_k(K_3,G)$ is the least positive integer $n$ such that every $k$-coloring of the edges of the complete graph on $n$ vertices contains either a rainbow triangle or a monochromatic copy of $G$. It turns out that $gr_k(K_3, G)$ behaves more nicely than the classical Ramsey number $r_k(G)$. However, finding exact values of $gr_k (K_3, G)$ is far from trivial. In this paper, we prove that $gr_k(K_3, C_9)= 4\cdot 2^k+1$ for all $k\ge1$. This new result provides partial evidence fo…
Let $G=(V,E)$ be a $τ$-critical graph with $τ(G)=t$. Erdős and Gallai proved that $|V|\leq 2t$ and the bound $|E|\leq {t+1\choose 2}$ was obtained by Erdős, Hajnal and Moon. We give here the sharp combined bound $|E|+|V|\leq {t+2\choose 2}$ and find all graphs with equality.
Given a graph $H$, the $k$-colored Gallai Ramsey number $gr_{k}(K_{3} : H)$ is defined to be the minimum integer $n$ such that every $k$-coloring of the edges of the complete graph on $n$ vertices contains either a rainbow triangle or a monochromatic copy of $H$. Fox et al. [J. Fox, A. Grinshpun, and J. Pach. The Erd{\H o}s-Hajnal conjecture for rainbow triangles. J. Combin. Theory Ser. B, 111:75-125, 2015.] conjectured the value of the Gallai Ramsey numbers for complete graphs. Recently, this conjecture has been verified for the first open case, when $H = K_{4}$. In this paper we attack the n…
F. Escalante and T. Gallai studied in the seventies the structure of different kind of separations and cuts between a vertex pair in a (possibly infinite) graph. One of their results is that if there is a finite separation, then the optimal (i.e. minimal sized) separations form a finite distributive lattice with respect to a natural partial order. Furthermore, any finite distributive lattice can be represented this way. If there is no finite separation then cardinality is a too rough measure to capture being 'optimal'. Menger's theorem provides a structural characterization of optimality if th…
Let $\mathcal{F}$ be a family of graphs. The Turán number $ex(n;\mathcal{F})$ is defined to be the maximum number of edges in a graph of order $n$ that is $\mathcal{F}$-free. In 1959, Erdős and Gallai determined the Turán number of $M_{k+1}$ (a matching of size $k+1$) as follows: \[ ex(n;M_{k+1})= \max\left\{\binom{2k+1}{2},\binom{n}{2}-\binom{n-k}{2}\right\}. \] Since then, there has been a lot of research on Turán number of linear forests. A linear forest is a graph whose connected components are all paths or isolated vertices. Let $\mathcal{L}_{n,k}$ be the family of all linear forests of o…
An Erdős-Ko-Rado result for some principal series representations
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学学院国家重点实验室）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
We prove that every 3-coloring of the edges of the complete graph on n vertices without a rainbow triangle contains a set of order Omega(n^{1/3}log^2 n) which uses at most two colors, and this bound is tight up to a constant factor. This verifies a conjecture of Hajnal which is a case of the multicolor generalization of the well-known Erdős-Hajnal conjecture. We further establish a generalization of this result. For fixed positive integers s and r with s at most r, we determine a constant c_{r,s} such that the following holds. Every r-coloring of the edges of the complete graph on n vertices w…
The extension of an $r$-uniform hypergraph $G$ is obtained from it by adding for every pair of vertices of $G$, which is not covered by an edge in $G$, an extra edge containing this pair and $(r-2)$ new vertices. In this paper we determine the Turán number of the extension of an $r$-graph consisting of two vertex-disjoint edges, settling a conjecture of Hefetz and Keevash, who previously determined this Turán number for $r=3$. As the key ingredient of the proof we show that the Lagrangian of intersecting $r$-graphs is maximized by principally intersecting $r$-graphs for $r \geq 4$.
Bidirected graphs generalize directed and undirected graphs in that edges are oriented locally at every node. The natural notion of the degree of a node that takes into account (local) orientations is that of net-degree. In this paper, we extend the following four topics from (un)directed graphs to bidirected graphs: - Erdős-Gallai-type results: characterization of net-degree sequences, - Havel-Hakimi-type results: complete sets of degree-preserving operations, - Extremal degree sequences: characterization of uniquely realizable sequences, and - Enumerative aspects: counting formulas for net-d…
A transversal set of a graph $G$ is a set of vertices incident to all edges of $G$. The transversal number of $G$, denoted by $τ(G)$, is the minimum cardinality of a transversal set of $G$. A simple graph $G$ with no isolated vertex is called $τ$-critical if $τ(G-e) < τ(G)$ for every edge $e\in E(G)$. For any $τ$-critical graph $G$ with $τ(G)=t$, it has been shown that $|V(G)|\le 2t$ by Erdős and Gallai and that $|E(G)|\le {t+1\choose 2}$ by Erdős, Hajnal and Moon. Most recently, it was extended by Gyárfás and Lehel to $|V(G)| + |E(G)|\le {t+2\choose 2}$. In this paper, we prove stronger resul…
For finite sequence $\underbar{\em d}$ of positive integers, we consider graphs that have $\underbar{\em d}$ as their list of vertex degrees, and bipartite graphs for which each part has $\underbar{\em d}$ as its list of vertex degrees. In particular, we make a connection between a result for bipartite graphs by Alon, Ben-Shimon and Krivelevich and a result of Zverovich and Zverovich for graphs, and we give an improvement of a result of Zverovich and Zverovich. We show that the bipartite graphs with vertex degree sequences $(\underbar{\em d},\underbar{\em d}\,)$ are in one to one correspondenc…
For many types of graphs, criteria have been discovered that give necessary and sufficient conditions for an integer sequence to be the degree sequence of such a graph. These criteria tend to take the form of a set of inequalities, and in the case of the Erdős-Gallai criterion (for simple undirected graphs) and the Gale-Ryser criterion (for bipartite graphs), it has been shown that the number of inequalities that must be checked can be reduced significantly. We show that similar reductions hold for the corresponding criteria for many other types of graphs, including bipartite r-multigraphs, bi…
Perhaps the very first elementary exercise one encounters in graph theory is the result that any graph on at least two vertices must have at least two vertices with the same degree. There are various ways in which this result can be non-trivially generalised. For example, one can interpret this result as saying that in any graph $G$ on at least two vertices there is a set $B$ of at least two vertices such that the difference between the largest and the smallest degrees (in $G$) of the vertices of $B$ is zero. In this vein we make the following definition. For any $B\subset V(G)$, let the sprea…
The degree sequence of a graph is the sequence of the degrees of its vertices. If $π$ is a degree sequence of a graph $G$, then $G$ is a realization of $π$ and $G$ realizes $π$. Determining when a sequence of positive integers is realizable as a degree sequence of a simple graph has received much attention. One of the early results, by Erdös and Gallai, characterized degree sequences of graphs. The result was strengthened by Hakimi and Havel. Another generalization is derived by Cai et al. Hoogeveen and Sierksma listed seven criteria and gave a uniform proof. In addition, Gale and Ryser indepe…
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…
We consider the following fundamental realization problem of directed graphs. Given a sequence $S:={a_1 \choose b_1},\dots,{a_n \choose b_n}$ with $a_i,b_i\in \mathbb{Z}_0^+.$ Does there exist a digraph (no loops and no parallel arcs are allowed)$G=(V,A)$ with a labeled vertex set $V:=\{v_1,\dots,v_n\}$ such that for all $v_i \in V$ indegree and outdegree of $v_i$ match exactly the given numbers $a_i$ and $b_i$, respectively? There exist two known approaches solving this problem in polynomial running time. One first approach of Kleitman and Wang (1973) uses recursive algorithms to construct di…
