Finding Paths and Cycles in Graphs

Properly Coloured Cycles and Paths: Results and Open Problems

In this paper, we consider a number of results and seven conjectures on properly edge-coloured (PC) paths and cycles in edge-coloured multigraphs. We overview some known results and prove new ones. In particular, we consider a family of transformations of an edge-coloured multigraph $G$ into an ordinary graph that allow us to check the existence PC cycles and PC $(s,t)$-paths in $G$ and, if they exist, to find shortest ones among them. We raise a problem of finding the optimal transformation and consider a possible solution to the problem.

On Rainbow Cycles and Paths

In a properly edge colored graph, a subgraph using every color at most once is called rainbow. In this thesis, we study rainbow cycles and paths in proper edge colorings of complete graphs, and we prove that in every proper edge coloring of K_n, there is a rainbow path on (3/4-o(1))n vertices, improving on the previously best bound of (2n+1)/3 from Gyarfas and Mhalla. Similarly, a k-rainbow path in a proper edge coloring of K_n is a path using no color more than k times. We prove that in every proper edge coloring of K_n, there is a k-rainbow path on (1-2/(k+1)!)n vertices.

Finding $k$ Simple Shortest Paths and Cycles

The problem of finding multiple simple shortest paths in a weighted directed graph $G=(V,E)$ has many applications, and is considerably more difficult than the corresponding problem when cycles are allowed in the paths. Even for a single source-sink pair, it is known that two simple shortest paths cannot be found in time polynomially smaller than $n^3$ (where $n=|V|$) unless the All-Pairs Shortest Paths problem can be solved in a similar time bound. The latter is a well-known open problem in algorithm design. We consider the all-pairs version of the problem, and we give a new algorithm to find…

Choosability of a weighted path and free-choosability of a cycle

A graph $G$ with a list of colors $L(v)$ and weight $w(v)$ for each vertex $v$ is $(L,w)$-colorable if one can choose a subset of $w(v)$ colors from $L(v)$ for each vertex $v$, such that adjacent vertices receive disjoint color sets. In this paper, we give necessary and sufficient conditions for a weighted path to be $(L,w)$-colorable for some list assignments $L$. Furthermore, we solve the problem of the free-choosability of a cycle.

A general purpose algorithm for counting simple cycles and simple paths of any length

We describe a general purpose algorithm for counting simple cycles and simple paths of any length $\ell$ on a (weighted di)graph on $N$ vertices and $M$ edges, achieving a time complexity of $O\left(N+M+\big(\ell^ω+\ellΔ\big) |S_\ell|\right)$. In this expression, $|S_\ell|$ is the number of (weakly) connected induced subgraphs of $G$ on at most $\ell$ vertices, $Δ$ is the maximum degree of any vertex and $ω$ is the exponent of matrix multiplication. We compare the algorithm complexity both theoretically and experimentally with most of the existing algorithms for the same task. These comparison…

Contracting Bipartite Graphs to Paths and Cycles

Testing if a given graph $G$ contains the $k$-vertex path $P_k$ as a minor or as an induced minor is trivial for every fixed integer $k\geq 1$. However, the situation changes for the problem of checking if a graph can be modified into $P_k$ by using only edge contractions. In this case the problem is known to be NP-complete even if $k=4$. This led to an intensive investigation for testing contractibility on restricted graph classes. We focus on bipartite graphs. Heggernes, van 't Hof, Lévêque and Paul proved that the problem stays NP-complete for bipartite graphs if $k=6$. We strengthen their …

关于最长路径与最长圈相交的两个猜想

On two conjectures about the intersection of longest paths and cycles

本文研究k-连通图中最长圈与最长路径的相交猜想，证明最长圈公共顶点数下界并在特定参数范围证实Smith猜想，还通过归约关联两类猜想并在特定k范围验证Hippchen猜想。

Partitioning two-coloured complete multipartite graphs into monochromatic paths and cycles

We show that any complete $k$-partite graph $G$ on $n$ vertices, with $k \ge 3$, whose edges are two-coloured, can be covered with two vertex-disjoint monochromatic paths of distinct colours. We prove this under the necessary assumption that the largest partition class of $G$ contains at most $n/2$ vertices. This extends known results for complete and complete bipartite graphs. Secondly, we show that in the same situation, all but $o(n)$ vertices of the graph can be covered with two vertex-disjoint monochromatic cycles of distinct colours, if colourings close to a split colouring are excluded.…

On Hamiltonian alternating cycles and paths

We undertake a study on computing Hamiltonian alternating cycles and paths on bicolored point sets. This has been an intensively studied problem, not always with a solution, when the paths and cycles are also required to be plane. In this paper, we relax the constraint on the cycles and paths from being plane to being 1-plane, and deal with the same type of questions as those for the plane case, obtaining a remarkable variety of results. Among them, we prove that a 1-plane Hamiltonian alternating cycle on a bicolored point set in general position can always be obtained, and that when the point…

Enumerating Chemical Graphs with Two Disjoint Cycles Satisfying Given Path Frequency Specifications

Enumerating chemical graphs satisfying given constraints is a fundamental problem in mathematical and computational chemistry, and plays an essential part in a recently proposed framework for the inverse QSAR/QSPR. In this paper, constraints are given by feature vectors each of which consists of the frequencies of paths in a given set of paths. We consider the problem of enumerating chemical graphs that satisfy the path frequency constraints, which are given by a pair of feature vectors specifying upper and lower bounds of the frequency of each path. We design a branch-and-bound algorithm for …

On the independent subsets of powers of paths and cycles

In the first part of this work we provide a formula for the number of edges of the Hasse diagram of the independent subsets of the h-th power of a path ordered by inclusion. For h=1 such a value is the number of edges of a Fibonacci cube. We show that, in general, the number of edges of the diagram is obtained by convolution of a Fibonacci-like sequence with itself. In the second part we consider the case of cycles. We evaluate the number of edges of the Hasse diagram of the independent subsets of the h-th power of a cycle ordered by inclusion. For h=1, and n>1, such a value is the number of e…

Transversals of Longest Paths and Cycles

Let G be a graph of order n. Let lpt(G) be the minimum cardinality of a set X of vertices of G such that X intersects every longest path of G and define lct(G) analogously for cycles instead of paths. We prove that lpt(G) \leq ceiling(n/4-n^{2/3}/90), if G is connected, lct(G) \leq ceiling(n/3-n^{2/3}/36), if G is 2-connected, and \lpt(G) \leq 3, if G is a connected circular arc graph. Our bound on lct(G) improves an earlier result of Thomassen and our bound for circular arc graphs relates to an earlier statement of Balister \emph{et al.} the argument of which contains a gap. Furthermore, we p…

Physical Zero-knowledge Proofs for Flow Free, Hamiltonian Cycles, and Many-to-many k-disjoint Covering Paths

In this paper we describe protocols which use a standard deck of cards to provide a perfectly sound zero-knowledge proof for Hamiltonian cycles and Flow Free puzzles. The latter can easily be extended to provide a protocol for a zero-knowledge proof of many-to-many k-disjoint path coverings.

Broadcasts on Paths and Cycles

A broadcast on a graph $G=(V,E)$ is a function $f: V\longrightarrow \{0,\ldots,\operatorname{diam}(G)\}$ such that $f(v)\leq e\_G(v)$ for every vertex $v\in V$, where$\operatorname{diam}(G)$ denotes the diameter of $G$ and $e\_G(v)$ the eccentricity of $v$ in $G$. The cost of such a broadcast is then the value $\sum\_{v\in V}f(v)$.Various types of broadcast functions on graphs have been considered in the literature, in relation with domination, irredundence, independenceor packing, leading to the introduction of several broadcast numbers on graphs.In this paper, we determine these broadcast nu…

Generalized Fibonacci and Lucas cubes arising from powers of paths and cycles

The paper deals with some generalizations of Fibonacci and Lucas sequences, arising from powers of paths and cycles, respectively. In the first part of the work we provide a formula for the number of edges of the Hasse diagram of the independent sets of the h-th power of a path ordered by inclusion. For h=1 such a diagram is called a Fibonacci cube, and for h>1 we obtain a generalization of the Fibonacci cube. Consequently, we derive a generalized notion of Fibonacci sequence, called h-Fibonacci sequence. Then, we show that the number of edges of a generalized Fibonacci cube is obtained by con…