Solution of Vizing's Problem on Interchanges for Graphs with Maximum Degree 4 and Related Results

Let $G$ be a Class 1 graph with maximum degree $4$ and let $t\geq 5$ be an integer. We show that any proper $t$-edge coloring of $G$ can be transformed to any proper $4$-edge coloring of $G$ using only transformations on $2$-colored subgraphs (so-called interchanges). This settles the smallest previously unsolved case of a well-known problem of Vizing on interchanges, posed in 1965. Using our result we give an affirmative answer to a question of Mohar for two classes of graphs: we show that all proper $5$-edge colorings of a Class 1 graph with maximum degree 4 are Kempe equivalent, that is, can be transformed to each other by interchanges, and that all proper 7-edge colorings of a Class 2 graph with maximum degree 5 are Kempe equivalent

Some local--global phenomena in locally finite graphs

In this paper we present some results for a connected infinite graph $G$ with finite degrees where the properties of balls of small radii guarantee the existence of some Hamiltonian and connectivity properties of $G$. (For a vertex $w$ of a graph $G$ the ball of radius $r$ centered at $w$ is the subgraph of $G$ induced by the set $M_r(w)$ of vertices whose distance from $w$ does not exceed $r$). In particular, we prove that if every ball of radius 2 in $G$ is 2-connected and $G$ satisfies the condition $d_G(u)+d_G(v)\geq |M_2(w)|-1$ for each path $uwv$ in $G$, where $u$ and $v$ are non-adjacent vertices, then $G$ has a Hamiltonian curve, introduced by K\"undgen, Li and Thomassen (2017). Furthermore, we prove that if every ball of radius 1 in $G$ satisfies Ore's condition (1960) then all balls of any radius in $G$ are Hamiltonian.Comment: 18 pages, 6 figures; journal accepted versio

On the number of partial Steiner systems

We give a simple proof of the result of Grable on the asymptotics of the number of partial Steiner systems S(t,k,m)

Stable properties of graphs

Abstract For many properties P Bondy and Chvátal (1976) have found sufficient conditions such that if a graph G + uv has property P then G itself has property P. In this paper we will give a generalization that will improve ten of these conditions

Some localization theorems on hamiltonian circuits

Theorems on the localization of the conditions of G. A. Dirac (Proc. London Math. Soc. (3), 2 1952, 69–81), O. Ore (Amer. Math. Monthly, 67 1960, 55), and Geng-hua Fan (J. Combin. Theory Ser. B, 37 1984, 221–227) for a graph to be hamiltonian are obtained. It is proved, in particular, that a connected graph G on p ≥ 3 vertices is hamiltonian if d(u) ≥ | M3(u)|/2 for each vertex u in G, where M3(u) is the set of vertices v in G that are a distance at most three from u

Some panconnected and pancyclic properties of graphs with a local ore-type condition

Asratian and Khachatrian proved that a connected graphG of order at least 3 is hamiltonian ifd(u) + d(v) ≥ |N(u) ∪ N(v) ∪ N(w)| for any pathuwv withuv ∉ E(G), whereN(x) is the neighborhood of a vertexx. We prove that a graphG with this condition, which is not complete bipartite, has the following properties: a) For each pair of verticesx, y with distanced(x, y) ≥ 3 and for each integern, d(x, y) ≤ n ≤ |V(G)| − 1, there is anx − y path of lengthn.  (b)For each edgee which does not lie on a triangle and for eachn, 4 ≤ n ≤ |V(G)|, there is a cycle of lengthn containinge.  (c)Each vertex ofG lies on a cycle of every length from 4 to |V(G)|.  This implies thatG is vertex pancyclic if and only if each vertex ofG lies on a triangle