A graph is called traceable if it contains a Hamilton path, i.e., a path containing all its vertices. Let G be a graph on n vertices. We say that an induced subgraph of G is o−1-heavy if it contains two nonadjacent vertices which satisfy an Ore-type degree condition for traceability, i.e., with degree sum at least n−1 in G. A block-chain is a graph whose block graph is a path, i.e., it is either a P1, P2, or a 2-connected graph, or a graph with at least one cut vertex and exactly two end-blocks....

For a given graph G and a positive integer r the r-path graph, $P_r\left(G\right)$, has for vertices the set of all paths of length r in G. Two vertices are adjacent when the intersection of the corresponding paths forms a path of length r-1, and their union forms either a cycle or a path of length k+1 in G. Let $P_r^k\left(G\right)$ be the k-iteration of r-path graph operator on a connected graph G. Let H be a subgraph of $P_r^k\left(G\right)$. The k-history $P_r^{-k}\left(H\right)$ is a subgraph of G that is induced by all edges that take part in the recursive definition of...

Let $(H,r)$ be a fixed rooted digraph. The $(H,r)$-coloring problem is the problem of deciding for which rooted digraphs $(G,s)$ there is a homomorphism $f:G\to H$ which maps the vertex $s$ to the vertex $r$. Let $(H,r)$ be a rooted oriented path. In this case we characterize the nonexistence of such a homomorphism by the existence of a rooted oriented cycle $(C,q)$, which is homomorphic to $(G,s)$ but not homomorphic to $(H,r)$. Such a property of the digraph $(H,r)$ is called rooted cycle duality or $*$-cycle duality. This extends the analogical result for...

Bipartite graphs G = (L,R;E) and H = (L’,R’;E’) are bi-placeabe if there is a bijection f:L∪R→ L’∪R’ such that f(L) = L’ and f(u)f(v) ∉ E’ for every edge uv ∈ E. We prove that if G and H are two bipartite balanced graphs of order |G| = |H| = 2p ≥ 4 such that the sizes of G and H satisfy ||G|| ≤ 2p-3 and ||H|| ≤ 2p-2, and the maximum degree of H is at most 2, then G and H are bi-placeable, unless G and H is one of easily recognizable couples of graphs. This result implies easily that for integers...

In 1982 Laborde, Payan and Xuong [Independent sets and longest directed paths in digraphs, in: Graphs and other combinatorial topics (Prague, 1982) 173-177 (Teubner-Texte Math., 59 1983)] conjectured that every digraph has an independent detour transversal (IDT), i.e. an independent set which intersects every longest path. Havet [Stable set meeting every longest path, Discrete Math. 289 (2004) 169-173] showed that the conjecture holds for digraphs with independence number two. A digraph is p-deficient...

We present several results concerning the Laborde-Payan-Xuang conjecture stating that in every digraph there exists an independent set of vertices intersecting every longest path. The digraphs we consider are defined in terms of local semicompleteness and local transitivity. We also look at oriented graphs for which the length of a longest path does not exceed 4.

The paper concerns infinite paths (in particular, the maximum number of pairwise vertex-disjoint ones) in locally finite graphs and in spanning trees of such graphs.

Let $p$ be a prime. We assign to each positive number $k$ a digraph $G_p^k$ whose set of vertices is $\{1,2,...,p-1\}$ and there exists a directed edge from a vertex $a$ to a vertex $b$ if $a^k\equiv b(modp)$. In this paper we obtain a necessary and sufficient condition for $G_p^{k_1}\simeq G_p^{k_2}$.

Let D be a digraph. D is said to be an m-colored digraph if the arcs of D are colored with m colors. A path P in D is called monochromatic if all of its arcs are colored alike. Let D be an m-colored digraph. A set N ⊆ V(D) is said to be a kernel by monochromatic paths of D if it satisfies the following conditions: a) for every pair of different vertices u,v ∈ N there is no monochromatic directed path between them; and b) for every vertex x ∈ V(D)-N there is a vertex n ∈ N such that there is an xn-monochromatic...

For a digraph D, V (D) and A(D) will denote the sets of vertices and arcs of D respectively. In an arc-colored digraph, a subset K of V(D) is said to be kernel by monochromatic paths (mp-kernel) if (1) for any two different vertices x, y in N there is no monochromatic directed path between them (N is mp-independent) and (2) for each vertex u in V (D) N there exists v ∈ N such that there is a monochromatic directed path from u to v in D (N is mp-absorbent). If every arc in D has a different color,...