Total interval numbers of complete r-partite graphs

AbstractA multiple-interval representation of a graph G is a mapping f which assigns to each vertex of G a union of intervals on the real line so that two distinct vertices u and v are adjacent if and only if f(u)∩f(v)≠∅. We study the total interval number of G, defined asI(G)=min∑v∈V#f(v):fisamultiple-intervalrepresentationofG,where #f(v) is the minimum number of intervals whose union is f(v). We give bounds on the total interval numbers of complete r-partite graphs. Exact values are also determined for several cases

Properties of Catlin's reduced graphs and supereulerian graphs

A graph $G$ is called collapsible if for every even subset $R\subseteq V(G)$, there is a spanning connected subgraph $H$ of $G$ such that $R$ is the set of vertices of odd degree in $H$. A graph is the reduction of $G$ if it is obtained from $G$ by contracting all the nontrivial collapsible subgraphs. A graph is reduced if it has no nontrivial collapsible subgraphs. In this paper, we first prove a few results on the properties of reduced graphs. As an application, for 3-edge-connected graphs $G$ of order $n$ with $d(u)+d(v)\ge 2(n/p-1)$ for any $uv\in E(G)$ where $p>0$ are given, we show how such graphs change if they have no spanning Eulerian subgraphs when $p$ is increased from $p=1$ to 10 then to $15$

The Total Interval Number of a Graph

Coordinated Science Laboratory changed its name from Control Systems LaboratoryOffice of Naval Research / N00014-85-K-0570Ope

The Total Interval Number of a Graph

123 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1988.An interval representation (or simply representation) R of a graph G is a collection of finite sets $\{R(\nu):\nu \in V(G)\}$ of closed bounded intervals so that $u \leftrightarrow \nu$ if and only if there exist \theta\sb{u} \in R(u), \theta\sb{\nu} \in R(\nu) with \theta\sb{u} \cap \theta\sb{\nu} \not= \emptyset. The size of a representation is the number of intervals in the entire collection.The total interval number of G is the size of the smallest representation of G and is denoted I(G). This thesis studies I by proving best possible upper bounds for several classes of graphs. For some classes, the bounds are in terms of n, the number of vertices and for some classes, the bounds are in terms of m, the number of edges. The main result is that for planar graphs, $I(G) \leq 2n(G) - 3$.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD

The Total Interval Number of a Graph

Coordinated Science Laboratory changed its name from Control Systems LaboratoryOffice of Naval Research / N00014-85-K-0570Ope