A Size Bound for Hamilton Cycles
Every graph of size $q$ (the number of edges) and minimum degree $δ$ is hamiltonian if $q\leδ^2+δ-1$. The result is sharp.
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arXiv subjects
Zh. G. Nikoghosyan
Publications and source records attributed to Zh. G. Nikoghosyan.
Graph Invariants and Large Cycles - a Catalog of Pure Links
Graph invariants provide a powerful analytical tool for investigation of abstract structures of graphs. They, combined in convenient relations, carry global and general information about a graph and its various substructures such as cycle structures, factors, matchings, colorings, coverings, and so on, whose discovery is the primary problem of graph theory. The major goal of this paper is to catalogue all pure relations between basic invariants of a graph and its large cycle structures, namely Hamilton, longest and dominating cycles and some their generalizations. Basic graph invariants and pure relations allow to focus on results having no forerunners. These simplest kind of "ancestors" form a source from which nearly all possible hamiltonian results can be developed further by various additional new ideas, generalizations, extensions, restrictions and structural limitations, as well as helping researchers to make clear and simple imagination about "complicated" developmental mechanisms in the area.
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A Lower Bound for the Circumference Involving Connectivity
Let $G$ be a graph, $C$ a longest cycle in $G$ and $\overline{p}$, $\overline{c}$ the lengths of a longest path and a longest cycle in $G\backslash C$, respectively. Almost all lower bounds for the circumference base on a standard procedure: choose an initial cycle $C_0$ in $G$ and try to enlarge it via structures of $G\backslash C_0$ and connections between $C_0$ and $G\backslash C_0$ closely related to $\overline{p}$, $\overline{c}$ and connectivity $κ$. Actually, each lower bound obtained in result of this procedure, somehow or is related to $κ$, $\overline{p}$, $\overline{c}$ but in forms of various particular values of $κ$, $\overline{p}$, $\overline{c}$ and the major problem is to involve these invariants into such bounds as parameters. In this paper we present a lower bound for the circumference involving $δ$, $κ$ and $\overline{c}$ and increasing with $δ$, $κ$ and $\overline{c}$.
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Large cycles in 4-connected graphs
Every 4-connected graph $G$ with minimum degree $δ$ and connectivity $κ$ either contains a cycle of length at least $4δ-κ-4$ or every longest cycle in $G$ is a dominating cycle.
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On the circumference, connectivity and dominating cycles
Every 4-connected graph with minimum degree $δ$ and connectivity $κ$ either has a cycle of length at least $4δ-2κ$ or has a dominating cycle.
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On Longest Cycle $C$ of a graph $G$ via Structures of $G-C$
Two sharp lower bounds for the length of a longest cycle $C$ of a graph $G$ are presented in terms of the lengths of a longest path and a longest cycle of $G-C$, denoted by $\overline{p}$ and $\overline{c}$, respectively, combined with minimum degree $δ$: (1) $|C|\geq(\overline{p}+2)(δ-\overline{p})$ and (2) $|C|\geq(\overline{c}+1)(δ-\overline{c}+1)$.
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Long cycles in graphs through fragments
Four basic Dirac-type sufficient conditions for a graph $G$ to be hamiltonian are known involving order $n$, minimum degree $δ$, connectivity $κ$ and independence number $α$ of $G$: (1) $δ\geq n/2$ (Dirac); (2) $κ\geq 2$ and $δ\geq (n+κ)/3$ (by the author); (3) $κ\geq 2$ and $δ\geq \max\lbrace (n+2)/3,α\rbrace$ (Nash-Williams); (4) $κ\geq 3$ and $δ\geq \max\lbrace (n+2κ)/4,α\rbrace$ (by the author). In this paper we prove the reverse version of (4) concerning the circumference $c$ of $G$ and completing the list of reverse versions of (1)-(4): (R1) if $κ\geq 2$, then $c\geq\min\lbrace n,2δ\rbrace$ (Dirac); (R2) if $κ\geq 3$, then $c\geq\min\lbrace n,3δ-κ\rbrace$ (by the author); (R3) if $κ\geq 3$ and $δ\geq α$, then $c\geq\min\lbrace n,3δ-3\rbrace$ (Voss and Zuluaga); (R4) if $κ\geq 4$ and $δ\geq α$, then $c\geq\min\lbrace n,4δ-2κ\rbrace$. To prove (R4), we present four more general results centered around a lower bound $c\geq 4δ-2κ$ under four alternative conditions in terms of fragments. A subset $X$ of $V(G)$ is called a fragment of $G$ if $N(X)$ is a minimum cut-set and $V(G)-(X\cup N(X))\neq\emptyset$.
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