Thomas Lam, Jacques Verstraete
We show that any n-vertex graph without even cycles of length at most 2k has at most 1/2(n^{1 + 1/k}) + O(n) edges, and polarity graphs of generalized polygons show that this is asymptotically tight when k = 2,3,5.
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In this note, we prove that every non-complete $(k+1)$-critical graph contains cycles of all lengths modulo $k$, where $k=4,5$.
Using some combinatorial techniques, in this note, it is proved that if $α\geq 0.28866$, then any digraph on $n$ vertices with minimum outdegree at least $αn$ contains a directed cycle of length at most 4.
Whitney's Broken-cycle Theorem states the chromatic polynomial of a graph as a sum over special edge subsets. We give a definition of cycles in hypergraphs that preserves the statement of the theorem there.
In this note we obtain upper bounds on the number of hyperedges in 3-uniform hypergraphs not containing a Berge cycle of given odd length. We improve the bound given by Füredi and Özkahya in 2017. The result follows from a more general theorem. We also obtain some new results for Berge cliques.
If $G$ is a more than one tough graph on $n$ vertices with $δ\ge \frac{n}{2}-a$ for a given $a>0$ and $n$ is large enough then $G$ is hamiltonian.
Let $G$ be a graph of radius $r$ and diameter $d$ with $d\leq 2r-2$. We give a new proof that $G$ contains a cycle of length at least $4r-2d$, i.e. for its circumference it holds $c(G)\geq 4r-2d$.
Balogh, Csaba, Jing and Pluhár recently determined the minimum degree threshold that ensures a $2$-coloured graph $G$ contains a Hamilton cycle of significant colour bias (i.e., a Hamilton cycle that contains significantly more than half of its edges in one colour). In this short note we extend this result, determining the corresponding threshold for $r$-colourings.
The strong cycle double cover conjecture states that for every circuit $C$ of a bridgeless cubic graph $G$, there is a cycle double cover of $G$ which contains $C$. We conjecture that there is even a 5-cycle double cover $S$ of $G$ which contains $C$, i.e. $C$ is a subgraph of one of the five 2-regular subgraphs of $S$. We prove a necessary and sufficient condition for a 2-regular subgraph to be contained in a 5-cycle double cover of $G$.
Two new sufficient conditions for generalized cycles (including Hamilton and dominating cycles as special cases) in an arbitrary k-connected graph (k=1,2,...) are derived, which prove the truth of Bondy's (1980) famous conjecture for some variants significantly improving the result expected by the given hypothesis. Similarly, two new lower bounds for the circumference (the length of a longest cycle) are established for the reverse hypothesis proposed by Jung (2001).
Let $G$ be an edge-colored graph. A heterochromatic cycle of $G$ is one in which every two edges have different colors. For a vertex $v\in V(G)$, let $CN(v)$ denote the set of colors which are assigned to the edges incident to $v$. In this note we prove that $G$ contains a heterochromatic cycle of length 4 if $G$ has $n\geq 60$ vertices and $|CN(u)\cup CN(v)|\geq n-1$ for every pair of vertices $u$ and $v$ of $G$. This extends a result of Broersma et al. on the existence of heterochromatic cycles of length 3 or 4.
We find the threshold for the existence of a collection of edge disjoint copies of $K_r$ that form a cyclic structure and span all vertices of $G_{n,p}$. We use a recent result of Riordan to give a two line proof of the main result.
We conjecture that a 2-connected graph $G$ of order $n$, in which $d(x)+d(y)\geq n-k$ for every pair of non-adjacent vertices $x$ and $y$, contains a cycle of length $n-k$ ($k<n/2$), unless $G$ is bipartite and $n-k$ is odd. This generalizes to long cycles a well-known degree sum condition for hamiltonicity of Ore. The conjecture is shown to hold for $k=1$.
Let $\mathcal{C}_{n,k}$ be the family of all cacti with $k$ cycles and $n\geq4$ vertices. In the present note, the element of the class $\mathcal{C}_{n,k}$ having minimum augmented Zagreb index ($AZI$) is characterized. Moreover, some structural properties of the graph(s) having maximum $AZI$ value over the collection $\mathcal{C}_{n,0}$, are also reported.
Let $L_{c,n}$ denote the size of the longest cycle in $G(n,{c}/{n})$, $c>1$ constant. We show that there exists a continuous function $f(c)$ such that $ L_{c,n}/n \to f(c)$ a.s. for $c\geq 20$, thus extending a result of the author and Frieze to smaller values of $c$. Thereafter, for $c\geq 20$, we determine the limit of the probability that $G(n,c/n)$ contains cycles of every length between the length of its shortest and its longest cycles as $n\to \infty$.
Borodin et al figured out a gap of the paper published at J. Combinatorial Theory Ser. B (Vol.96 (2006) 958--963), and gave a new proof with the similar technique. The purpose of this note is to fix the gap by slightly revising the definition of special faces, and adding a few lines of explanation in the proofs (new added text are all in black font).
In 1992, Manoussakis conjectured that a strongly 2-connected digraph $D$ on $n$ vertices is hamiltonian if for every two distinct pairs of independent vertices $x,y$ and $w,z$ we have $d(x)+d(y)+d(w)+d(z)\geq 4n-3$. In this note we show that $D$ has a Hamilton path, which gives an affirmative evidence supporting this conjecture.
Let $G$ be an $n$-vertex graph obtained by adding chords to a cycle of length $n$. Markström asked for the maximum number of edges in $G$ if there are no two cycles in $G$ with the same length. A simple counting argument shows that such a graph can have at most $n + \sqrt{2n} +1 $ edges. Using difference sets in $\mathbb{Z}_n$, we show that for infinitely many $n$, there is an $n$-vertex Hamiltonian graph with $n + \sqrt{n - 3/4} - 3/2$ edges and no repeated cycle length.
A cycle C={v_1,v_2,....,v_1} in a tournament T is said to be even, if when walking along C, an even number of edges point in the wrong direction, that is, they are directed from v_{i+1} to v_i. In this short paper, we show that for every fixed even integer k >= 4, if close to half of the k-cycles in a tournament T are even, then T must be quasi-random. This resolves an open question raised in 1991 by Chung and Graham
We prove bijectively that the total number of cycles of all even permutations of $[n]=\{1,2,...,n\}$ and the total number of cycles of all odd permutations of $[n]$ differ by $(-1)^n(n-2)!$, which was stated as an open problem by Miklós Bóna. We also prove bijectively the following more general identity: $$\sum_{i=1}^n c(n,i)\cdot i \cdot (-k)^{i-1} = (-1)^k k! (n-k-1)!,$$ where $c(n,i)$ denotes the number of permutations of $[n]$ with $i$ cycles.
Note on Long Directed Cycles in Eulerian Digraphs
本研究针对欧拉有向图中长有向环的长度下界问题，改进了Huang等人此前得出的下界结果，将其从$\sqrt{d}$提升至$\sqrt{2d}-3/2.$。
A weighted digraph is a digraph such that every arc is assigned a nonnegative number, called the weight of the arc. The weighted outdegree of a vertex $v$ in a weighted digraph $D$ is the sum of the weights of the arcs with $v$ as their tail, and the weight of a directed cycle $C$ in $D$ is the sum of the weights of the arcs of $C$. In this note we prove that if every vertex of a weighted digraph $D$ with order $n$ has weighted outdegree at least 1, then there exists a directed cycle in $D$ with weight at least $1/\log_2 n$. This proves a conjecture of Bollobás and Scott up to a constant facto…
Let $D$ be a strong digraph on $n\geq 4$ vertices. In [3, Discrete Applied Math., 95 (1999) 77-87)], J. Bang-Jensen, Y. Guo and A. Yeo proved the following theorem: if (*) $d(x)+d(y)\geq 2n-1$ and $min \{d^+(x)+ d^-(y),d^-(x)+ d^+(y)\}\geq n-1$ for every pair of non-adjacent vertices $x, y$ with a common in-neighbour or a common out-neighbour, then $D$ is hamiltonian. In this note we show that: if $D$ is not directed cycle and satisfies the condition (*), then $D$ contains a cycle of length $n-1$ or $n-2$.
Paul Erdős and László Lovász established by means of an example that there exists a maximal intersecting family of $k-$sets with approximately $(e-1)k!$ blocks. László Lovász conjectured that their example is best known example which has the maximum number of blocks. Later it was disproved. But the quest for such examples remain valid till this date. In this short note, by computing transversal size of a certain series of uniform intersecting families constructed over the cycle graph, we provide an example which has more than $(\frac{k}{2})^{k-1}$ (approximately) blocks.
A Note on Hamiltonian Cycles in Digraphs with Large Degrees
本文证明了阶为 $n$ 的 2-强有向图在满足特定顶点度数下界及存在经过特定顶点的长圈条件时必为哈密顿图。
Note on Disjoint Cycles in Multipartite Tournaments
本文针对无三角形多部竞赛图和3部竞赛图验证了Bermond-Thomassen关于顶点不相交有向圈的猜想，并进一步刻画了最小出度至少为$2k-1$且任意$k$个不相交圈等长的3部竞赛图。
A note on long nontrivial cycle in Hamiltonian graphs
本文改进了Girão等人关于哈密顿图中存在长非平凡环的长度下界，从n-O(n^{4/5})提升至n-O(n^{2/3})，并证明该界在所用方法下是最优的。
Let $G$ be a graph of order $n$ with an edge-coloring $c$, and let $δ^c(G)$ denote the minimum color degree of $G$. A subgraph $F$ of $G$ is called rainbow if all edges of $F$ have pairwise distinct colors. There have been a lot results on rainbow cycles of edge-colored graphs. In this paper, we show that (i) if $δ^c(G)>\frac{3n-3}{4}$, then every vertex of $G$ is contained in a rainbow triangle; (ii) $δ^c(G)>\frac{3n}{4}$, then every vertex of $G$ is contained in a rainbow $C_4$; and (iii) if $G$ is complete, $n\geq 8k-18$ and $δ^c(G)>\frac{n-1}{2}+k$, then $G$ contains a rainbow cycle of len…
A note on cycles in cyclically $4$-edge-connected cubic planar graphs
研究通过删除两个相邻顶点得到的图H，证明当H的周长至少为k时，存在长度在k到3k/2之间的循环，并展示线图G中可避免特定顶点的循环长度。
Let ${\cal{C}}_1$ be the set of fundamental cycles of breadth-first-search trees in a graph $G$ and ${\cal{C}}_2$ the set of the sums of two cycles in ${\cal{C}}_1$. Then we show that $(1) {\cal{C}}={\cal{C}}_1\bigcup{\cal{C}}_2$ contains a shortest $Π$-twosided cycle in a $Π$-embedded graph $G$;$(2)$ $\cal{C}$ contains all the possible shortest even cycles in a graph $G$;$(3)$ If a shortest cycle in a graph $G$ is an odd cycle, then $\cal{C}$ contains all the shortest odd cycles in $G$. This implies the existence of a polynomially bounded algorithm to find a shortest $Π-$twosided cycle in an …
A note on the number of non-cycle components in a pseudo 2-factor of graphs
本文推广了图伪2-因子的非圈分支数量上界，证明每个图G存在非圈分支数至多为f(G)的伪2-因子，统一了Bekkai-Kouider与作者先前的2-因子存在性结果。
How many edges can a quadrilateral-free subgraph of a hypercube have? This question was raised by Paul Erdős about $27$ years ago. His conjecture that such a subgraph asymptotically has at most half the edges of a hypercube is still unresolved. Let $f(n,C_l)$ be the largest number of edges in a subgraph of a hypercube $Q_n$ containing no cycle of length $l$. It is known that $f(n, C_l) = o(|E(Q_n)|)$, when $l= 4k$, $k\geq 2$ and that $f(n, C_6) \geq \frac{1}{3} |E(Q_n)|$. It is an open question to determine $f(n, C_l)$ for $l=4k+2$, $k\geq 2$. Here, we give a general upper bound for $f(n,C_l)$…
Let $A$ be a set of natural numbers and let $S_{n,A}$ be the set of all permutations of $[n]=\{1,2,...,n\}$ with cycle lengths belonging to $A$. For $A(n)=A\cap [n]$, the limit $ρ=\lim_{n\to\infty}\mid A(n)\mid/n$ (if it esists) is usually called the density of set $A$. (Here $\mid B\mid$ stands for the cardinality of the set $B$.) Several studies show that the asymptotic behavior of the cardinality $\mid S_{n,A}\mid$, as $n\to\infty$, depends on the density $ρ$. It turns out that the asumption $ρ>0$ plays an essential role in the asymptotic analysis of $\mid S_{n,A}\mid$. Kolchin (1999) notic…
Let $k,a,b$ be positive integers with $a+b=k$. A $k$-uniform hypergraph is called an $(a,b)$-cycle if there is a partition $(A_0,B_0,A_1,B_1,\ldots,A_{t-1},B_{t-1})$ of the vertex set with $|A_i|=a$, $|B_i|=b$ such that $A_i\cup B_i$ and $B_i\cup A_{i+1}$ (subscripts module $t$) are edges for all $i=0,1,\ldots,t-1$. Let $\mathcal{H}$ be a $k$-uniform $n$-vertex hypergraph with $n\geq 5k$ and $n$ divisible by $k$. By applying the concentration inequality for intersections of a uniform hypergraph with a random matching developed by Frankl and Kupavskii, we show that if there exists $α\in (0,1)$ …
Note on the treewidth of graphs excluding a disjoint union of cycles as a minor
本文研究了不包含离散环联合作为小世界的图的树宽界限，证明了其与图顶点数和环数的关系，并得出最优结果。
Let $t>0$ be a real number and $G$ be a graph. We say $G$ is $t$-tough if for every cutset $S$ of $G$, the ratio of $|S|$ to the number of components of $G-S$ is at least $t$. The Toughness Conjecture of Chvátal, stating that there exists a constant $t_0$ such that every $t_0$-tough graph with at least three vertices is hamiltonian, is still open in general. For any given integer $k\ge 1$, a graph $G$ is $(P_2\cup kP_1)$ free if $G$ does not contain the disjoint union of $P_2$ and $k$ isolated vertices as an induced subgraph. In this note, we show that every 4-tough and $2k$-connected $(P_2\cu…
Let G be a graph. It was proved that if G is a planar graph without {4, 6, 7}-cycles and without two 5-cycles sharing exactly one edge, then G 3-colorable. We observed that the proof of this result is not correct.
Hamidoune's connectivity results for hierarchical Cayley digraphs are extended to Cayley coset digraphs and thus to arbitrary vertex transitive digraphs. It is shown that if a Cayley coset digraph can be hierarchically decomposed in a certain way, then it is optimally vertex connected. The results are obtained by extending the methods used by Hamidoune. They are used to show that cycle-prefix graphs are optimally vertex connected. This implies that cycle-prefix graphs have good fault tolerance properties.
A note on tree-cycle Ramsey numbers
本文围绕树 - 圈拉姆齐数展开，Burr等人曾询问满足特定条件的最小函数\(f(m)\)，前人不断改进其上限，本文用不同方法证明\(f(m)\leq2m - 4\)，并提出\(f(m)=\lceil (2m - 1)/3\rceil\)的猜想。
Two Hamilton paths in $K_n$ are separated by a cycle of length $k$ if their union contains such a cycle. For small fixed values of $k$ we bound the asymptotics of the maximum cardinality of a family of Hamilton paths in $K_n$ such that any pair of paths in the family is separated by a cycle of length $k.$
Let $R_k(H;K_m)$ be the smallest number $N$ such that every coloring of the edges of $K_{N}$ with $k+1$ colors has either a monochromatic $H$ in color $i$ for some $1\leqslant i\leqslant k$, or a monochromatic $K_{m}$ in color $k+1$. In this short note, we study the lower bound for $R_k(H;K_m)$ when $H$ is $C_5$ or $C_7$, respectively. We show that \begin{equation*} R_{k}(C_5;K_m)=Ω(m^{\frac{3k}{8}+1}/(\log{m})^{\frac{3k}{8}+1}), \end{equation*} and \begin{equation*} R_{k}(C_7;K_m)=Ω(m^{\frac{2k}{9}+1}/(\log{m})^{\frac{2k}{9}+1}), \end{equation*} for fixed positive integer $k$ and $m\rightarro…
The Shortest Cycle Cover Conjecture of Alon and Tarsi asserts that the edges of every bridgeless graph with $m$ edges can be covered by cycles of total length at most $7m/5=1.400m$. We show that every cubic bridgeless graph has a cycle cover of total length at most $34m/21\approx 1.619m$ and every bridgeless graph with minimum degree three has a cycle cover of total length at most $44m/27\approx 1.630m$.
We prove that a complete bipartite graph can be decomposed into cycles of arbitrary specified lengths provided that the obvious necessary conditions are satisfied, the length of each cycle is at most the size of the smallest part, and the longest cycle is at most three times as long as the second longest. We then use this result to obtain results on incomplete even cycle systems with a hole and on decompositions of complete multipartite graphs into cycles of uniform even length.
The density, denoted by $κ(n,r)$, of permutations having no cycles of length less than $r+1$ in a symmetric group $\mathrm{S}_n$ is explored. New asymptotic formulas for $κ(n,r)$ are obtained using the saddle-point method when $5\leq r< n$ and $n\to\infty$.
Two cycles are {\em adjacent} if they have an edge in common. Suppose that $G$ is a planar graph, for any two adjacent cycles $C_{1}$ and $C_{2}$, we have $|C_{1}| + |C_{2}| \geq 11$, in particular, when $|C_{1}| = 5$, $|C_{2}| \geq 7$. We show that the graph $G$ is 3-colorable.
An $(L,d)^*$-coloring is a mapping $ϕ$ that assigns a color $ϕ(v)\in L(v)$ to each vertex $v\in V(G)$ such that at most $d$ neighbors of $v$ receive colore $ϕ(v)$. A graph is called $(m,d)^*$-choosable, if $G$ admits an $(L,d)^*$-coloring for every list assignment $L$ with $|L(v)|\geq m$ for all $v\in V(G)$. In this note, it is proved that every toroidal graph, which contains no adjacent triangles and contains no 6-cycles and $l$-cycles for some $l \in \{5,7\}$, is $(3,1)^*$-choosable.
The Game of Cycles is an impartial game on a planar graph that was introduced by Francis Su. In this short note we address some questions that have been raised on the game, and raise some further questions.
In this short note we determine the maximum number, over all $n$-vertex graphs $G$, of orientations of $G$ containing no strongly connected cycle $C_{2k+1}$. This answers a part of a recent question of Araujo, Botler and Mota.
Finding a Hamilton graph from simple connected graphs is an important problem in discrete mathematics and computer science. Grinberg Theorem is a well-known necessary condition for planar Hamilton graphs. It divides a plane into two parts: inside and outside faces. The sum of inside faces in a Hamilton graph is a Hamilton cycle. In this paper, using a basis of the cycle space to represent a graph and by the Inclusion-Exclusion Principle, we derive the equality with respect to the inside faces that can be also obtained from Grinberg Theorem. By further investigating the cycle structure of insid…
A graph is {\em near-bipartite} if its vertex set can be partitioned into an independent set and a set that induces a forest. It is clear that near-bipartite graphs are $3$-colorable. In this note, we show that planar graphs without cycles of lengths in $\{4, 6, 8\}$ are near-bipartite.
A Note on ID-Colorings and Symmetric Colorings of Cycles
本文针对阶为素数的圈，利用染色对称性建立了判定红白染色是否为ID-染色的准则，将Kono关于路径的结果推广到圈上。
A note on the threshold numbers of cycles
本文确定了圈的阈值数：$C_3$为1，$C_4$为2，$C_n(n\ge5)$为4，通过构造和证明给出了精确结果。
OpenAI's proof of the Cycle Double Cover Theorem
尝试阐明OpenAI对圈双覆盖猜想的证明，虽无新见解，但更易读，为理解其证明提供便利。
In this short note, we give a new sufficient condition for the existence of long cycles in graphs involving Fan-type degree condition and neighborhood intersection.
We determine the maximum number of edges in a $K_4$-minor-free $n$-vertex graph of girth $g$, when $g = 5$ or $g$ is even. We argue that there are many different $n$-vertex extremal graphs, if $n$ is even and $g$ is odd.
Let $P(n,m)$ be a graph chosen uniformly at random from the class of all planar graphs on vertex set $\{1, \ldots, n\}$ with $m=m(n)$ edges. We study the cycle and block structure of $P(n,m)$ when $m\sim n/2$. More precisely, we determine the asymptotic order of the length of the longest and shortest cycle in $P(n,m)$ in the critical range when $m=n/2+o(n)$. In addition, we describe the block structure of $P(n,m)$ in the weakly supercritical regime when $n^{2/3}\ll m-n/2\ll n$.
We present various results on multiplying cycles in the symmetric group. Our first result is a generalisation of the following theorem of Boccara (1980): the number of ways of writing an odd permutation in the symmetric group on $n$ symbols as a product of an $n$-cycle and an $n-1$-cycle is independent of the permutation chosen. We give a number of different approaches of our generalisation. One partial proof uses an inductive method which we also apply to other problems. In particular, we give a formula for the distribution of the number of cycles over all products of cycles of fixed lengths.…
For any graph $G$ with $a,b\in V(G)$, a shortest path reconfiguration graph can be formed with respect to $a$ and $b$; we denote such a graph as $S(G,a,b)$. The vertex set of $S(G,a,b)$ is the set of all shortest paths from $a$ to $b$ in $G$ while two vertices $U,W$ in $V(S(G,a,b))$ are adjacent if and only if the vertex sets of the paths that represent $U$ and $W$ differ in exactly one vertex. In a recent paper [Asplund et al., \textit{Reconfiguration graphs of shortest paths}, Discrete Mathematics \textbf{341} (2018), no. 10, 2938--2948], it was shown that shortest path graphs with girth fiv…
In this short note we prove that for every $k\in \mathbb{N}$ there is a $t_k\in\mathbb{N}$ such that for every digraph $G$ there are either $k$ edge-disjoint directed cycles in $G$ or a set $X$ of at most $t_k$ edges such that $G-X$ contains no directed cycle.
We characterise digraphs of directed treewidth one in terms of forbidden butterfly minors. Moreover, we show that there is a linear relation between the hypertree-width of the dual of the cycle hypergraph of D, i. e. the hypergraph with vertices V (D) where every hyperedge corresponds to a directed cycle in D, and the directed treewidth of D. Based on this we show that a digraph has directed treewidth one if and only if its cycle hypergraph is a hypertree.
A matching $M$ in a graph $G$ is uniquely restricted if no other matching in $G$ covers the same set of vertices. We prove that any connected subcubic graph with $n$ vertices and girth at least $5$ contains a uniquely restricted matching of size at least $(n-1) / 3$ except for two exceptional cubic graphs of order $14$ and $20$.
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