Let G be a graph on n ≥ 3 vertices. A graph G is almost distance-hereditary if each connected induced subgraph H of G has the property dH(x, y) ≤ dG(x, y) + 1 for any pair of vertices x, y ∈ V(H). Adopting the terminology introduced by Broersma et al. and Čada, a graph G is called 1-heavy if at least one of the end vertices of each induced subgraph of G isomorphic to K1,3 (a claw) has degree at least n/2, and is called claw-heavy if each claw of G has a pair of end vertices with degree sum at least...

By a hamiltonian coloring of a connected graph $G$ of order $n\ge 1$ we mean a mapping $c$ of $V\left(G\right)$ into the set of all positive integers such that $|c\left(x\right)-c\left(y\right)|\ge n-1-D_G(x,y)$ (where $D_G(x,y)$ denotes the length of a longest $x-y$ path in $G$) for all distinct $x,y\in G$. In this paper we study hamiltonian colorings of non-hamiltonian connected graphs with long cycles, mainly of connected graphs of order $n\ge 5$ with circumference $n-2$.

The family of 5-valent polyhedral graphs whose faces are all triangles or 3s-gons, s ≥ 9, is shown to contain non-hamiltonian graphs and to have a shortness exponent smaller than one.

Following a problem posed by Lovász in 1969, it is believed that every finite connected vertex-transitive graph has a Hamilton path. This is shown here to be true for cubic Cayley graphs arising from finite groups having a $(2,s,3)$-presentation, that is, for groups $G=\langle a,b\mid a^2=1,b^s=1,{\left(ab\right)}^3=1,\cdots \rangle$ generated by an involution $a$ and an element $b$ of order $s\ge 3$ such that their product $ab$ has order $3$. More precisely, it is shown that the Cayley graph $X=Cay(G,\{a,b,b^{-1}\})$ has a Hamilton cycle when $\left|G\right|$ (and thus $s$) is congruent to 2 modulo 4, and has a long cycle missing...

Let G be a graph on n vertices. A vertex of G with degree at least n/2 is called a heavy vertex, and a cycle of G which contains all the heavy vertices of G is called a heavy cycle. In this note, we characterize graphs which contain no heavy cycles. For a given graph H, we say that G is H-heavy if every induced subgraph of G isomorphic to H contains two nonadjacent vertices with degree sum at least n. We find all the connected graphs S such that a 2-connected graph G being S-heavy implies any longest...

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...