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 embedded graph and thus solves an open problem of B.Mohar and C.Thomassen[2,pp112]
The varietal hypercube $VQ_n$ is a variant of the hypercube $Q_n$ and has better properties than $Q_n$ with the same number of edges and vertices. This paper shows that every edge of $VQ_n$ is contained in cycles of every length from 4 to $2^n$ except 5, and every pair of vertices with distance $d$ is connected by paths of every length from $d$ to $2^n-1$ except 2 and 4 if $d=1$.
The existence of a primitive element of $GF(q)$ with certain properties is used to prove that all cycles that could theoretically be embedded in $AG(2,q)$ and $PG(2,q)$ can, in fact, be embedded there (i.e. these planes are `pancyclic'). We also study embeddings of wheel and gear graphs in arbitrary projective planes.
We define and study embeddings of cycles in finite affine and projective planes. We show that for all $k$, $3\le k\le q^2$, a $k$-cycle can be embedded in any affine plane of order $q$. We also prove a similar result for finite projective planes: for all $k$, $3\le k\le q^2+q+1$, a $k$-cycle can be embedded in any projective plane of order $q$.
We derive improved upper bounds on the number of crossing-free straight-edge spanning cycles (also known as Hamiltonian tours and simple polygonizations) that can be embedded over any specific set of $N$ points in the plane. More specifically, we bound the ratio between the number of spanning cycles (or perfect matchings) that can be embedded over a point set and the number of triangulations that can be embedded over it. The respective bounds are $O(1.8181^N)$ for cycles and $O(1.1067^N)$ for matchings. These imply a new upper bound of $O(54.543^N)$ on the number of crossing-free straight-edge…
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.
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 $k.$
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.
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.
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.
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.
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…
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 $20$.
Let $G$ be a non-bipartite graph which does not contain any odd cycle of length at most $2k+1$. In this paper, we determine the maximum $Q$-index of $G$ if its order is fixed, and the corresponding extremal graph is uniquely characterized. Moreover, if the size of $G$ is given, the maximum $Q$-index of $G$ and the unique extremal graph are also proved.
We study graphs on $n$ vertices which have $2n-2$ edges and no proper induced subgraphs of minimum degree $3$. Erdős, Faudree, Gyárfás, and Schelp conjectured that such graphs always have cycles of lengths $3,4,5,\dots, C(n)$ for some function $C(n)$ tending to infinity. We disprove this conjecture, resolve a related problem about leaf-to-leaf path lengths in trees, and characterize graphs with $n$ vertices and $2n-2$ edges, containing no proper subgraph of minimum degree $3$.
A graph is outerplanar if it can be embedded in a plane such that all vertices lie on its outer face. The outerplanar Turán number of a given graph $H$, denoted by ${\rm ex}_{\mathcal{OP}}(n,H)$, is the maximum number of edges over all outerplanar graphs on $n$ vertices which do not contain a copy of $H$. In this paper, the outerplanar Turán numbers of cycles and paths are completely determined.
Properly colored cycles in edge-colored graphs are closely related to directed cycles in oriented graphs. As an analogy of the well-known Caccetta-Häggkvist Conjecture, we study the existence of properly colored cycles of bounded length in an edge-colored graph. We first prove that for all integers $s$ and $t$ with $t\geq s\geq2$, every edge-colored graph $G$ with no properly colored $K_{s,t}$ contains a spanning subgraph $H$ which admits an orientation $D$ such that every directed cycle in $D$ is a properly colored cycle in $G$. Using this result, we show that for $r\geq4$, if the Caccetta-Hä…
Felsner introduced a cycle reversal, namely the `flip' reversal, for α-orientations (i.e., each vertex admits a prescribed out-degree) of a graph G embedded on the plane and further proved that the set of all the α-orientations of G carries a distributive lattice with respect to the flip reversals. In this paper, we give an explicit formula for the minimum number of flips needed to transform one α-orientation into another for graphs embedded on the plane or sphere, respectively.
A list assignment of a graph $G$ is a function $L$ that assigns a list $L(v)$ of colors to each vertex $v\in V(G)$. An $(L,d)^*$-coloring is a mapping $π$ that assigns a color $π(v)\in L(v)$ to each vertex $v\in V(G)$ so that at most $d$ neighbors of $v$ receive color $π(v)$. A graph $G$ is said to be $(k,d)^*$-choosable if it admits an $(L,d)^*$-coloring for every list assignment $L$ with $|L(v)|\ge k$ for all $v\in V(G)$. In 2001, Lih et al. \cite{LSWZ-01} proved that planar graphs without 4- and $l$-cycles are $(3,1)^*$-choosable, where $l\in \{5,6,7\}$. Later, Dong and Xu \cite{DX-09} prov…
In the theory of digraphs, the study of cycles is a subject of great importance and has given birth to a number of deep questions such as the Behzad-Chartrand-Wall conjecture (1970) and its generalization, the Caccetta-Häggkvist conjecture (1978). Despite a lot of interest and efforts, the progress on these remains slow and mostly restricted to the solution of some special cases. In this note, we prove these conjectures for digraphs with girth is at least as large as their minimum out-degree and without short even cycles. More generally, we prove that if a digraph has sufficiently large girth …
For a directed graph $G$ without loops or parallel edges, let $β(G)$ denote the size of the smallest feedback arc set, i.e., the smallest subset $X \subset E(G)$ such that $G \sm X$ has no directed cycles. Let $γ(G)$ be the number of unordered pairs of vertices of $G$ which are not adjacent. We prove that every directed graph whose shortest directed cycle has length at least $r \ge 4$ satisfies $β(G) \le cγ(G)/r^2$, where $c$ is an absolute constant. This is tight up to the constant factor and extends a result of Chudnovsky, Seymour, and Sullivan. This result can be also used to answer a quest…
In this paper we show that many snarks have shortest cycle covers of length $\frac{4}{3}m+c$ for a constant $c$, where $m$ is the number of edges in the graph, in agreement with the conjecture that all snarks have shortest cycle covers of length $\frac{4}{3}m+o(m)$. In particular we prove that graphs with perfect matching index at most 4 have cycle covers of length $\frac{4}{3}m$ and satisfy the $(1,2)$-covering conjecture of Zhang, and that graphs with large circumference have cycle covers of length close to $\frac{4}{3}m$. We also prove some results for graphs with low oddness and discuss th…
Consider the family of graphs without $ k $ node-disjoint odd cycles, where $ k $ is a constant. Determining the complexity of the stable set problem for such graphs $ G $ is a long-standing problem. We give a polynomial-time algorithm for the case that $ G $ can be further embedded in a (possibly non-orientable) surface of bounded genus. Moreover, we obtain polynomial-size extended formulations for the respective stable set polytopes. To this end, we show that $2$-sided odd cycles satisfy the Erdős-Pósa property in graphs embedded in a fixed surface. This extends the fact that odd cycles sati…
We show that every bridgeless cubic graph $G$ with $m$ edges has a cycle cover of length at most $1.6 m$. Moreover, if $G$ does not contain any intersecting circuits of length $5$, then $G$ has a cycle cover of length $212/135 \cdot m \approx 1.570 m$ and if $G$ contains no $5$-circuits, then it has a cycle cover of length at most $14/9 \cdot m \approx 1.556 m$. To prove our results, we show that each $2$-edge-connected cubic graph $G$ on $n$ vertices has a $2$-factor containing at most $n/10+f(G)$ circuits of length $5$, where the value of $f(G)$ only depends on the presence of several subgra…
A cycle cover of a graph is a collection of cycles such that each edge of the graph is contained in at least one of the cycles. The length of a cycle cover is the sum of all cycle lengths in the cover. We prove that every bridgeless cubic graph with $m$ edges has a cycle cover of length at most $212/135 \cdot m \ (\approx 1.570 m)$. Moreover, if the graph is cyclically $4$-edge-connected we obtain a cover of length at most $47/30 \cdot m \approx 1.567 m$.
We consider the number of crossings in a graph which is embedded randomly on a convex set of points. We give an estimate to the normal distribution in Kolmogorov distance which implies a convergence rate of order $n^{-1/2}$ for various families of graphs, including random chord diagrams or full cycles.
In this paper, we aim to introduce the group version of edge coloring and list edge coloring, and prove that all 2-degenerate graphs along with some planar graphs without adjacent short cycles is group $(Δ(G)+1)$-edge-choosable while some planar graphs with large girth and maximum degree is group $Δ(G)$-edge-choosable.
The saturation number of a graph $F$, written $\textup{sat}(n,F)$, is the minimum number of edges in an $n$-vertex $F$-saturated graph. One of the earliest results on saturation numbers is due to Erdős, Hajnal, and Moon who determined $\textup{sat}(n,K_r)$ for all $r \geq 3$. Since then, saturation numbers of various graphs and hypergraphs have been studied. Motivated by Alon and Shikhelman's generalized Turán function, Kritschgau et.\ al.\ defined $\textup{sat}(n,H,F)$ to be the minimum number of copies of $H$ in an $n$-vertex $F$-saturated graph. They proved, among other things, that $\textu…
The Caccetta-Häggkvist conjecture implies that for every integer $k\ge 1$, if $G$ is a bipartite digraph, with $n$ vertices in each part, and every vertex has out-degree more than $n/(k+1)$, then $G$ has a directed cycle of length at most $2k$. If true this is best possible, and we prove this for $k = 1,2,3,4,6$ and all $k\ge 224,539$. More generally, we conjecture that for every integer $k\ge 1$, and every pair of reals $α, β> 0$ with $kα+β>1$, if $G$ is a bipartite digraph with bipartition $(A,B)$, where every vertex in $A$ has out-degree at least $β|B|$, and every vertex in $B$ has out-degr…
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 $f(n,C_l)$…
A well-known result of Verstraëte \cite{V00} shows that for each integer $k\geq 2$ every graph $G$ with average degree at least $8k$ contains cycles of $k$ consecutive even lengths, the shortest of which is at most twice the radius of $G$. We establish two extensions of Verstraëte's result for linear cycles in linear $r$-uniform hypergraphs. We show that for any fixed integers $r\geq 3,k\geq 2$, there exist constants $c_1=c_1(r)$ and $c_2=c_2(r,k)$, such that every linear $r$-uniform hypergraph $G$ with average degree $d(G)\geq c_1 k$ contains linear cycles of $k$ consecutive even lengths, the…
In this paper we estimate the planar Turán number $\mathrm{ex}_\mathcal{P}(n,H)$ of some graphs $H$, i.e., the maximum number of edges in a planar graph $G$ of $n$ vertices not containing $H$ as a subgraph. We give a new, short proof when $H=C_5$, and study the cases when $G$ is bipartite or triangle-free and $H$ is a short even cycle. The proofs are mostly new applications or variants of the "contribution method" introduced by Ghosh, Győri, Martin, Paulos and Xiao in arXiv:2004.14094.
We study two extremal problems about subgraphs excluding a family $\F$ of graphs. i) Among all graphs with $m$ edges, what is the smallest size $f(m,\F)$ of a largest $\F$--free subgraph? ii) Among all graphs with minimum degree $δ$ and maximum degree $Δ$, what is the smallest minimum degree $h(δ,Δ,\F)$ of a spanning $\F$--free subgraph with largest minimum degree? These questions are easy to answer for families not containing any bipartite graph. We study the case where $\F$ is composed of all even cycles of length at most $2r$, $r\geq 2$. In this case, we give bounds on $f(m,\F)$ and $h(δ,Δ,…
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.
An oriented 3-graph consists of a family of triples (3-sets), each of which is given one of its two possible cyclic orientations. A cycle in an oriented 3-graph is a positive sum of some of the triples that gives weight zero to each 2-set. Our aim in this paper is to consider the following question: how large can the girth of an oriented 3-graph (on $n$ vertices) be? We show that there exist oriented 3-graphs whose shortest cycle has length $\frac{n^2}{2}(1+o(1))$: this is asymptotically best possible. We also show that there exist 3-tournaments whose shortest cycle has length $\frac{n^2}{3}(1…
The $book$ $embedding$ of a graph $G$ is to place the vertices of $G$ on the spine and draw the edges to the pages so that the edges in the same page do not cross with each other. The book embedding is $matching$ if the pages have maximum degree $1$. The $matching$ $book$ $thickness$ is the minimum number of pages in which graphs can be matching book embedded. In this paper, we show that the matching book thickness of the Cartesian product $K_p\Box C_q$of a complete graph $K_p$ and a cycle $C_q$ is equal to $Δ(K_p\Box C_q)+1$.
Online Ramsey game is played between Builder and Painter on an infinite board $K_{\mathbb N}$. In every round Builder selects an edge, then Painter colors it red or blue. Both know target graphs $H_1$ and $H_2$. Builder aims to create either a red copy of $H_1$ or a blue copy of $H_2$ in $K_{\mathbb N}$ as soon as possible, and Painter tries to prevent it. The online Ramsey number $\tilde{r}(H_1,H_2)$ is the minimum number of rounds such that the Builder wins. We study $\tilde{r}(C_k,C_n)$ where $k$ is fixed and $n$ is large. We show that $\tilde{r}(C_k,C_n)=2n+\mathcal O(k)$ for an absolute c…
It is well known that spectral Turán type problem is one of the most classical {problems} in graph theory. In this paper, we consider the spectral Turán type problem. Let $G$ be a graph and let $\mathcal{G}$ be a set of graphs, we say $G$ is \textit{$\mathcal{G}$-free} if $G$ does not contain any element of $\mathcal{G}$ as a subgraph. Denote by $λ_1$ and $λ_2$ the largest and the second largest eigenvalues of the adjacency matrix $A(G)$ of $G,$ respectively. In this paper we focus on the characterization of graphs without short odd cycles according to the adjacency eigenvalues of the graphs. …
We study large uniform random maps with one face whose genus grows linearly with the number of edges, which are a model of discrete hyperbolic geometry. In previous works, several hyperbolic geometric features have been investigated. In the present work, we study the number of short cycles in a uniform unicellular map of high genus, and we show that it converges to a Poisson distribution. As a corollary, we obtain the law of the systole of uniform unicellular maps in high genus. We also obtain the asymptotic distribution of the vertex degrees in such a map.
Recently, working on the Tanner graph which represents a low density parity check (LDPC) code becomes an interesting research subject. Finding the number of short cycles of Tanner graphs motivated Blake and Lin to investigate the multiplicity of cycles of length girth in bi-regular bipartite graphs, by using the spectrum and degree distribution of the graph. Although there were many algorithms to find the number of cycles, they preferred to investigate in a computational way. Dehghan and Banihashemi counted the number of cycles of length $g+2$ and $g+4,$ where $G$ is a bi-regular bipartite gra…
We study certain spaces of vertex functions on the Cayley graphs corresponding to N-fold products of the group of integers modulo m, where m=3, 4, or 5, that are invariant under the adjacency operator that maps a value at a given vertex to each of its neighbors. An application to spatio-spectral limiting, an analogue of time and band limiting, is also discussed.
For a weighted outerplanar graph, the set of lex short cycles is known to be a minimum cycle basis [Inf. Process. Lett. 110 (2010) 970-974 ]. In this work, we show that the set of lex short cycles is a minimum cycle basis in weighted partial 2-trees (graphs of treewidth two) which is a superclass of outerplanar graphs.
For any odd $t\ge 9$, we present a polynomial-time algorithm that solves the $3$-colouring problem, and finds a $3$-colouring if one exists, in $P_{t}$-free graphs of odd girth at least $t-2$. In particular, our algorithm works for $(P_9, C_3, C_5)$-free graphs, thus making progress towards determining the complexity of $3$-colouring in $P_t$-free graphs, which is open for $t\ge 8$.
