Monochromatic Paths Research Articles

$$\varPi $$ Π -Kernels in Digraphs

Let $$D=(V(D), A(D))$$D=(V(D),A(D)) be a digraph, $$DP(D)$$DP(D) be the set of directed paths of $$D$$D and let $$\varPi $$? be a subset of $$DP(D)$$DP(D). A subset $$S\subseteq V(D)$$S⊆V(D) will be called $$\varPi $$?-independent if for any pair $$\{x, y\} \subseteq S$${x,y}⊆S, there is no $$xy$$xy-path nor $$yx$$yx-path in $$\varPi $$?; and will be called $$\varPi $$?-absorbing if for every $$x\in V(D)\setminus S$$x?V(D)\S there is $$y\in S$$y?S such that there is an $$xy$$xy-path in $$\varPi $$?. A set $$S\subseteq V(D)$$S⊆V(D) will be called a $$\varPi $$?-kernel if $$S$$S is $$\varPi $$?-independent and $$\varPi $$?-absorbing. This concept generalize several "kernel notions" like kernel or kernel by monochromatic paths, among others. In this paper we present some sufficient conditions for the existence of $$\varPi $$?-kernels.

Partitioning 2-edge-colored graphs by monochromatic paths and cycles

We present results on partitioning the vertices of $2$-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of S\'ark\"ozy: the vertex set of every $2$-edge-colored graph can be partitioned into at most $2\alpha(G)$ monochromatic cycles, where $\alpha(G)$ denotes the independence number of $G$. Another direction, emerged recently from a conjecture of Schelp, is to consider colorings of graphs with given minimum degree. We prove that apart from $o(|V(G)|)$ vertices, the vertex set of any $2$-edge-colored graph $G$ with minimum degree at least $(1+\eps){3|V(G)|\over 4}$ can be covered by the vertices of two vertex disjoint monochromatic cycles of distinct colors. Finally, under the assumption that $\overline{G}$ does not contain a fixed bipartite graph $H$, we show that in every $2$-edge-coloring of $G$, $|V(G)|-c(H)$ vertices can be covered by two vertex disjoint paths of different colors, where $c(H)$ is a constant depending only on $H$. In particular, we prove that $c(C_4)=1$, which is best possible.

An Extension of Richardson’s Theorem in m-Colored Digraphs

A digraph $$D$$D is said to be an $$m$$m-coloured digraph if its arcs are coloured with $$m$$m colors. A directed path is called monochromatic if all of its arcs are coloured alike. A set $$N \subseteq $$N⊆ V ($$D$$D) of vertices of $$D$$D is said to be a kernel by monochromatic paths of the $$m$$m-coloured digraph $$D$$D if it satisfies the two following properties : (1) for any two different vertices $$x$$x, $$y$$y$$\in $$? N there is no monochromatic directed path between them, and (2) for each vertex $$u$$u$$\in $$? V($$D$$D)$$\setminus N$$\N there exists a $$uv$$uv-monochromatic directed path, for some $$v$$v$$\in N$$?N. Let $$D$$D be an arc-coloured digraph. In 2009 Galeana-Sanchez introduced the concept of color-class digraph of $$D$$D, denoted by $$\fancyscript{C}_C$$CC($$D$$D), as follows: the vertices of the color-class digraph are the colors represented in the arcs of $$D$$D, and ($$i$$i, $$j$$j) $$\in $$?$$A$$A($$\fancyscript{C}_C$$CC($$D$$D)) if and only if there exist two arcs namely ($$u$$u, $$v$$v) and ($$v$$v, $$w$$w) in $$D$$D such that ($$u$$u, $$v$$v) has color $$i$$i and ($$v$$v, $$w$$w) has color $$j$$j. Galeana-Sanchez introduced the concept of color-class digraph mainly to prove that if $$D$$D is a fnite arc-coloured digraph such that $$\fancyscript{C}_C$$CC($$D$$D) is a bipartite digraph, then D has a kernel by monochromatic paths. In this paper we will see some results related to the above result, which show the existence of kernels by monochromatic paths in arc-coloured digraphs. In particular, we will prove the following generalization of the previous result: Let $$D$$D be an $$m$$m-coloured digraph and $$\fancyscript{C}_C$$CC($$D$$D) its color-class digraph. If $$\fancyscript{C}_C$$CC($$D$$D) has no cycles of odd length at least 3, then $$D$$D has a kernel by monochromatic paths. Finally we will prove Richardson's Theorem as a direct consequence of the previous result.

Partitioning edge-coloured complete graphs into monochromatic cycles and paths

A conjecture of Erdős, Gyárfás, and Pyber says that in any edge-colouring of a complete graph with r colours, it is possible to cover all the vertices with r vertex-disjoint monochromatic cycles. So far, this conjecture has been proven only for r=2. In this paper we show that in fact this conjecture is false for all r⩾3. In contrast to this, we show that in any edge-colouring of a complete graph with three colours, it is possible to cover all the vertices with three vertex-disjoint monochromatic paths, proving a particular case of a conjecture due to Gyárfás. As an intermediate result we show that in any edge-colouring of the complete graph with the colours red and blue, it is possible to cover all the vertices with a red path, and a disjoint blue balanced complete bipartite graph.

Restricted domination in arc-colored digraphs

Let H = (V(H), A(H)) be a digraph possibly with loops and D = (V(D), A(D)) a digraph whose arcs are colored with the vertices of H (this is what we call an H-colored digraph); i.e. there exists a function c: A(D) → V(H); for an arc of D, f = (u, v) ∊ A(D), we call c(f) = c(u, v) the color of f. A directed walk (directed path) P = (u0, u1,…, un) in D will be called an H-walk (H-path) whenever (c(u0, u1), c(u1, u2),…, c(un-2, un-1), c(un-1, un)) is a directed walk (directed path) in H. We introduce the concept of H-kernel N, as a generalization of the two properties that define a kernel (Recall that a kernel N of a digraph D is a set of vertices N ⊆ V (D) which is independent and for each x ∊ V(D) — N, there exists an xN-arc in D). A set N ⊆ V(D) is called H-independent whenever for every two different vertices x, y ∊ N there is no H-path between them, and N is called H-absorbent whenever for each x ∊ V (D) — N there exists a vertex y ∊ N and an xy-H-path in D. The set N ⊆ V(D) will be called H-kernel if and only if it is H-independent and H-absorbent.This new concept generalizes the concepts of kernel, kernel by monochromatic paths and kernel by alternating paths.In this paper we show sufficient conditions for an infinite digraph to have an H-kernel.

On Panchromatic Digraphs and the Panchromatic Number

Let D = (V, A) be a simple finite digraph, and let $${\pi(D)}$$ ? ( D ) , the panchromatic number of D, be the maximum number of colours k such that for each (effective, or onto) colouring of its arcs $${\varsigma : A \to [k]}$$ ? : A ? [ k ] a monochromatic path kernel N ? V, as introduced in Galeana-Sanchez (Discrete Math 184: 87---99, 1998), exists. It is not hard to see that D has a kernel--in the sense of Von Neumann and Morgenstern (Theory of games and economic behaviour. Princeton University Press, Princeton, 1944)--if and only if $${\pi(D) = |A|}$$ ? ( D ) = | A | . In this note this invariant is introduced and some of its structural bounds are studied. For example, the celebrated theorem of Sands et al. (J Comb Theory Ser 33: 271---275, 1982), in terms of this invariant, settles that $${\pi(D) \geq 2}$$ ? ( D ) ? 2 . It will be proved that $$\pi(D) < |A| \iff \pi(D) < \min \left\{2\sqrt{\chi(D)}, \chi(L(D)), \theta(D) + \max {\text d}_c(K_i) + 1\right\},$$ ? ( D ) < | A | ? ? ( D ) < min 2 ? ( D ) , ? ( L ( D ) ) , ? ( D ) + max d c ( K i ) + 1 , where $${\chi(\cdot)}$$ ? ( · ) denotes the chromatic number, $${L(\cdot)}$$ L ( · ) denotes the line digraph, $${\theta(\cdot)}$$ ? ( · ) denotes the minimum partition into complete graphs of the underlying graph and $${\text{d}_c(\cdot)}$$ d c ( · ) denotes the dichromatic number. We also introduce the notion of a panchromatic digraph which is a digraph D such that for every k ≤ |A| and every k-colouring of its arcs, it has a monochromatic path kernel. Some classes of panchromatic digraphs are further characterised.

On hypercube labellings and antipodal monochromatic paths

A labelling of the n-dimensional hypercube Hn is a mapping that assigns value 0 or 1 to each edge of Hn. A labelling is antipodal if antipodal edges of Hn get assigned different values. It has been conjectured that, if Hn,n≥2, is given a labelling that is antipodal, then there exists a pair of antipodal vertices joined by a monochromatic path. This conjecture has been verified by hand for n≤5. In this paper, we verify the conjecture in the case where the labelling is simple in the sense that no square xyzt in Hn has value 0 assigned to xy,zt and value 1 assigned to yz,tx, even if the given labelling is not antipodal. The proof is based on a new property of (not necessarily antipodal) simple labellings of Hn. We also exhibit a large class of simple labellings that thus satisfy the conjecture. Finally, we conjecture that, even if the given labelling is not antipodal, there is always a path joining antipodal vertices that switches labels at most once, which implies the original conjecture. We establish this new conjecture for Hn,n≤5 as well.

Monochromatic Path and Cycle Partitions in Hypergraphs

Here we address the problem to partition edge colored hypergraphs by monochromatic paths and cycles generalizing a well-known similar problem for graphs.We show that $r$-colored $r$-uniform complete hypergraphs can be partitioned into monochromatic Berge-paths of distinct colors. Also, apart from $2k-5$ vertices, $2$-colored $k$-uniform hypergraphs can be partitioned into two monochromatic loose paths.In general, we prove that in any $r$-coloring of a $k$-uniform hypergraph there is a partition of the vertex set intomonochromatic loose cycles such that their number depends only on $r$ and $k$.

γ-cycles and transivity by monochromatic paths in arc-coloured digraphs

We call the digraph D an m-coloured digraph if its arcs are coloured with m colours. If D is an m-coloured digraph and a ∈ A(D), colour(a) will denote the colour has been used on a. A path (or a cycle) is called monochromatic if all of its arcs are coloured alike. A γ-cycle in D is a sequence of vertices, say γ = (u0, u1, . . . , un), such that ui 6= uj if i 6= j and for every i ∈ {0, 1, . . . , n} there is a uiui+1-monochromatic path in D and there is no ui+1ui-monochromatic path in D (the indices of the vertices will be takenmodn+1). A setN ⊆ V (D) is said to be a kernel by monochromatic paths if it satisfies the following two conditions: (i) for every pair of different vertices u, v ∈ N there is no monochromatic path between them and; (ii) for every vertex x ∈ V (D) \N there is a vertex y ∈ N such that there is an xy-monochromatic path. Let D be a finite m-coloured digraph. Suppose that {C1, C2} is a partition of C, the set of colours of D, and Di will be the spanning subdigraph of D such that A(Di) = {a ∈ A(D) | colour(a) ∈ Ci}. In this paper, we give some sufficient conditions for the existence of a kernel by monochromatic paths in a digraph with the structure mentioned above. In particular 494 E. Casas-Bautista, H. Galeana Sanchez and R. Rojas-Monroy we obtain an extension of the original result by B. Sands, N. Sauer and R. Woodrow that asserts: Every 2-coloured digraph has a kernel by monochromatic paths. Also, we extend other results obtained before where it is proved that under some conditions an m-coloured digraph has no γ-cycles.

A graph-based model of object recognition self-learning

Abstract In this paper, we study the object recognition self-learning for robots. In particular, we consider the self-learning during solution of typical tasks. We propose a graph-based model for self-learning. This model is based on the problem of monochromatic path for given set of weights. We prove that the problem is NP-complete. We consider an approach to solve the problem. This approach is based on an explicit reduction from the problem to the satisfiability problem.

Tournaments with kernels by monochromatic paths

In this paper we prove the existence of kernels by monochromatic paths in m-coloured tournaments in which every cyclic tournament of order 3 is atmost 2-coloured in addition to other restrictions on the colouring ofcertain subdigraphs. We point out that in all previous results on kernelsby monochromatic paths in arc coloured tournaments, certain smallsubstructures are required to be monochromatic or monochromatic with atmost one exception, whereas here we allow up to three colours in two smallsubstructures.

H-Kernels in Infinite Digraphs

Let H be a digraph possibly with loops and D a digraph (possibly infinite) without loops whose arcs are coloured with the vertices of H (D is an H-coloured digraph). V(D) and A(D) will denote the sets of vertices and arcs of D respectively. A directed walk or a directed path W in D is an H-walk or an H-path if and only if the consecutive colors encountered on W form a directed walk in H. A set $${N \subseteq V(D}$$ ) is an H-kernel if for every pair of different vertices in N there is no H-path between them, and for every vertex $${u \in V(D) \setminus N}$$ there exists an H-path in D from u to N. Linek and Sands introduced the concept of H-walk and this concept was later used by several authors. In particular, Galeana-Sanchez and Delgado-Escalante used the concept of H-walk in order to introduce the concept of H-kernel, which generalizes the concepts of kernel and kernel by monochromatic paths. Let D be an arc-coloured digraph. In 2009 Galeana-Sanchez introduced the concept of color-class digraph of D, denoted by $${\fancyscript{C}_C(D),}$$ as follows: the vertices of the color-class digraph are the colors represented in the arcs of D, and $${(i, j) \in A(\fancyscript{C}_C(D))}$$ if and only if there exist two arcs namely (u, v) and (v, w) in D such that (u, v) has color i and (v, w) has color j. Since V $${(\fancyscript{C}_C(D)) \subseteq {\rm V}(H)}$$ , the main question is: What structural properties of $${\fancyscript{C}_C(D),}$$ with respect to H, imply that D has an H-kernel? Suppose that D has no infinite outward H-path. In this paper we prove that if $${\fancyscript{C}_C(D) \subseteq H}$$ , then D has an H-kernel. We also prove that if there exists a partition (V 1, V 2) of V $${(\fancyscript{C}_C(D))}$$ such that: (1) $${\fancyscript{C}_C(D)[V_i] \subseteq H[V_i]}$$ for each i $${\in}$$ {1,2}, (2) if $${(u, v) \in A(\fancyscript{C}_C(D))}$$ for some $${u \in V_i}$$ and for some $${v \in V_j}$$ , with i ? j and $${i, j \in}$$ {1,2}, then $${(u, v) \notin A(H),}$$ and (3) D has no V i -colored infinite outward H-path for each i $${\in}$$ {1,2}. Then D has an H-kernel. Several previous results are generalized.

Partitioning 3-coloured complete graphs into three monochromatic paths

In this paper we show that in any edge-colouring of the complete graph by three colours, it is possible to cover all the vertices by three disjoint monochromatic paths. This solves a particular case of a conjecture of Gyarfas. As an intermediate result, we show that in any edge colouring of the complete graph by two colours, it is possible to cover all the vertices by a monochromatic path and a disjoint monochromatic balanced complete bipartite graph.

Design and Application of the Embedded System Based Online Monitoring SiO&lt;sub&gt;4&lt;/sub&gt;&lt;sup&gt;-2&lt;/sup&gt; Instrument

For the requirement to control the content of traces SiO4-2 in the water and vapor in coal-fired power plants. By thie method of using spectrophotometry, the multichannel online monitoring SiO4-2 instrument is designed based on embedded system according to the Lambert-Beer theorem. The entire system is managed by the embedded real-time operating system uC/OS, based on the industrial grade embedded microprocessor AT9140800. The system satisfactorily realized the detecting of traces SiO4-2 in high purity water by the plunger pumps, cold light source monochromatic double paths optical system, “reversal reagent dosing” method. The experiments and field application show that the system has stably function, high intelligence grade, high test precision and good reproducibility, so it really meets the requirements of detecting traces SiO4-2 in different industries.

Colorful monochromatic connectivity

An edge-coloring of a connected graph is monochromatically-connecting if there is a monochromatic path joining any two vertices. How “colorful” can a monochromatically-connecting coloring be? Let mc(G) denote the maximum number of colors used in a monochromatically-connecting coloring of a graph G. We prove some nontrivial upper and lower bounds for mc(G) and relate it to other graph parameters such as the chromatic number, the connectivity, the maximum degree, and the diameter.

Kernels by monochromatic paths and the color-class digraph

An m-coloured digraph is a digraph whose arcs are coloured with m colors. A directed path is monochromatic when its arcs are coloured alike. A set S ⊆ V (D) is a kernel by monochromatic paths whenever the two following conditions hold: 1. For any x, y ∈ S, x 6= y, there is no monochromatic directed path between them. 2. For each z ∈ (V (D)− S) there exists a zS-monochromatic directed path In this paper it is introduced the concept of color-class digraph to prove that if D is an m-coloured strongly connected finite digraph such that: (i) Every closed directed walk has an even number of color changes, (ii) Every directed walk starting and ending with the same color has an even number of color changes, then D has a kernel by monochromatic paths. This result generalizes a classical result by Sands, Sauer and Woodrow which asserts that any 2-coloured digraph has a kernel by monochromatic paths, in case that the digraph D be a strongly connected digraph. 2000 Mathematics Subject classification: 05C20.

On monochromatic paths and bicolored subdigraphs in arc-colored tournaments

On monochromatic paths and bicolored subdigraphs in arc-colored tournaments

Monochromatic cycles and monochromatic paths in arc-colored digraphs

Monochromatic cycles and monochromatic paths in arc-colored digraphs

Kernels by monochromatic paths in digraphs with covering number 2

We call the digraph D an k -colored digraph if the arcs of D are colored with k colors. A subdigraph H of D is called monochromatic if all of its arcs are colored alike. A set N ⊆ V ( D ) is said to be a kernel by monochromatic paths if it satisfies the following two conditions: (i) for every pair of different vertices u , v ∈ N , there is no monochromatic directed path between them, and (ii) for every vertex x ∈ ( V ( D ) ∖ N ) , there is a vertex y ∈ N such that there is an x y -monochromatic directed path. In this paper, we prove that if D is an k -colored digraph that can be partitioned into two vertex-disjoint transitive tournaments such that every directed cycle of length 3 , 4 or 5 is monochromatic, then D has a kernel by monochromatic paths. This result gives a positive answer (for this family of digraphs) of the following question, which has motivated many results in monochromatic kernel theory: Is there a natural number l such that if a digraph D is k - colored so that every directed cycle of length at most l is monochromatic, then D has a kernel by monochromatic paths?

Monochromatic paths and monochromatic sets of arcs in quasi-transitive digraphs

We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours. A directed path is called monochromatic if all of its arcs are coloured alike. A set N of vertices of D is called a kernel by monochromatic paths if for every pair of vertices of N there is no monochromatic path between them and for every vertex v / ∈ N there is a monochromatic path from v to N . We denote by A(u) the set of arcs of D that have u as the initial vertex. We prove that if D is an m-coloured 3-quasitransitive digraph such that for every vertex u of D, A(u) is monochromatic and D satisfies some colouring conditions over one subdigraph of D of order 3 and two subdigraphs of D of order 4, then D has a kernel by monochromatic paths.