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### A note on the interval number of a graph

Paul Erdös, Douglas B. West
1985 Discrete Mathematics
Acknowledgment We thank the referees for strengthening Theorem 2 for m 23.  ...  In this note, we apply counting arguments and results on graph decomposition to obtain inequalities concerning the interval number of a graph G.  ...  Such a description of G is called a t-representation of 6. Extremal results on the interval number of a graph have given us upper bounds on i(G) in terms of other graph parameters; see Table 1 .  ...

### Reconstruction of Interval Graphs [chapter]

Masashi Kiyomi, Toshiki Saitoh, Ryuhei Uehara
2009 Lecture Notes in Computer Science
Since the number of interval graphs that can be obtained from an interval graph by adding a vertex and edges incident to it can be exponentially large, developing polynomial time algorithms for LEGITIMATE  ...  The graph reconstruction conjecture is a long-standing open problem in graph theory.  ...  Note that the number of maximal cliques in an n-vertex interval graph is at most n (see [4] ). Lemma 6. All the compact interval representations of an interval graph have the same length.  ...

### The total interval number of a graph, I: Fundamental classes

Thomas M. Kratzke, Douglas B. West
1993 Discrete Mathematics
West, The total interval number of a graph, I: Fundamental classes, Discrete Mathematics 118 (1993) 145-156.  ...  The total interual number I(G) is the minimum of the total number of intervals used in any such representation of G.  ...  Then i(G) = min,,,F max, t VCG, #f(a) is called the interval number of G; the interval graphs are the graphs with interval number 1.  ...

### Reconstruction of interval graphs

Masashi Kiyomi, Toshiki Saitoh, Ryuhei Uehara
2010 Theoretical Computer Science
Since the number of interval graphs that can be obtained from an interval graph by adding a vertex and edges incident to it can be exponentially large, developing polynomial time algorithms for LEGITIMATE  ...  The graph reconstruction conjecture is a long-standing open problem in graph theory.  ...  Note that the number of maximal cliques in an n-vertex interval graph is at most n (see [4] ). Lemma 6. All the compact interval representations of an interval graph have the same length.  ...

### On a special class of boxicity 2 graphs [article]

Sujoy Kumar Bhore, Dibyayan Chakraborty, Sandip Das, Sagnik Sen
2016 arXiv   pre-print
We define and study a class of graphs, called 2-stab interval graphs (2SIG), with boxicity 2 which properly contains the class of interval graphs.  ...  It turns out that for these kind of graphs, the chromatic number of any of its induced subgraphs is bounded by twice of its (induced subgraph) clique number.  ...  Note that G[V 1 ] and G[V 2 ] are unit interval graphs. We prove the statement using induction on clique number ω(G). Note that the theorem is true for graphs G with ω(G) = 2 by Lemma 4.8.  ...

### On Interval Non-Edge-Colorable Eulerian Multigraphs [article]

Petros A. Petrosyan
2013 arXiv   pre-print
In this note, we show that all Eulerian multigraphs with an odd number of edges have no interval coloring.  ...  ,t is called an interval t-coloring if all colors are used, and the colors of edges incident to any vertex of G are distinct and form an interval of integers.  ...  For a graph G, by L(G) we denote the line graph of the graph G. We also need a classical result on Eulerian multigraphs.  ...

### Folkmusic and computers

Brian Beddington
1968 African Music: Journal of the African Music Society
If the final note is flat of the octave, the vps number will be less than twice that of the tonic, the factors will be unequal, and the note will fall short of the last position on the graph, as in graph  ...  The process is repeated for each subsequent note until the octave is reached, giving a result of nought on subtraction since the vps number of the octave is exactly twice that of the tonic, and both have  ...  The interval between a note and the tonic is found by adding together all the intervals of the intermediate notes.  ...

### Coloring Clean and K 4-Free Circle Graphs [chapter]

Alexandr V. Kostochka, Kevin G. Milans
2012 Thirty Essays on Geometric Graph Theory
A circle graph is the intersection graph of chords drawn in a circle. The best known general upper bound on the chromatic number of circle graphs with clique number k is 50 · 2 k .  ...  Based on this result we prove that the chromatic number of every circle graph with clique number at most 3 is at most 38.  ...  The above-mentioned complexity results on circle graphs make interesting upper bounds on the chromatic number of circle graphs in terms of their clique number, especially if the proofs yield polynomial  ...

### Coloring Fuzzy Circular Interval Graphs

Friedrich Eisenbrand, Martin Niemeier
2009 Electronic Notes in Discrete Mathematics
The algorithm reduces the problem to the case of circular interval graphs, then making use of a coloring algorithm by Gijswijt.  ...  We provide a polynomial time combinatorial algorithm that computes the weighted coloring number and the corresponding colorings for fuzzy circular interval graphs.  ...  If both endpoints have at least one node, we have a fuzzy pair. Let be the number of fuzzy pairs. Note that ≤ |V (G)| as fuzzy pairs are disjoint.  ...

### The harmonious coloring problem is NP-complete for interval and permutation graphs

Katerina Asdre, Kyriaki Ioannidou, Stavros D. Nikolopoulos
2007 Discrete Applied Mathematics
Given a simple graph G, a harmonious coloring of G is a proper vertex coloring such that each pair of colors appears together on at most one edge.  ...  Extending previous work on the NP-completeness of the harmonious coloring problem when restricted to the class of disconnected graphs which are simultaneously cographs and interval graphs, we prove that  ...  One can easily verify that G is an interval graph. A clique can be represented as a number of intervals that share at least one point in common.  ...

### The interval number of a planar graph: Three intervals suffice

Edward R Scheinerman, Douglas B West
1983 Journal of combinatorial theory. Series B (Print)
The interval number i(G) of a graph G is the smallest number t such that G has a t-interval representation. It is proved that i(G) < 3 whenever G is planar and that this bound is the best possible.  ...  Suppose each vertex of a graph G is assigned a subset of the real line consisting of at most t closed intervals.  ...  He noted that the interval number of a graph is at most one more than its arboricity, which is the minimum number of spanning forests needed to partition its edges.  ...

### Coloring fuzzy circular interval graphs

Friedrich Eisenbrand, Martin Niemeier
2012 European journal of combinatorics (Print)
The algorithm reduces the problem to the case of circular interval graphs, then making use of a coloring algorithm by Gijswijt.  ...  We provide a polynomial time combinatorial algorithm that computes the weighted coloring number and the corresponding colorings for fuzzy circular interval graphs.  ...  If both endpoints have at least one node, we have a fuzzy pair. Let be the number of fuzzy pairs. Note that ≤ |V (G)| as fuzzy pairs are disjoint.  ...

### A Note on Online Colouring Problems in Overlap Graphs and Their Complements [chapter]

Marc Demange, Martin Olsen
2018 Lecture Notes in Computer Science
Our method is based on a partition of the overlap graph into permutation graphs, leading to a competitive-preserving reduction of the problem in overlap graphs to the same problem in permutation graphs  ...  An instance is a system of time intervals presented in non-decreasing order of the left endpoints.  ...  Acknowledgments The authors would like to thank anonymous referees for their helpful comments.  ...

### A register allocation framework based on hierarchical cyclic interval graphs [chapter]

Laurie J. Hendren, Guang R. Gao, Erik R. Altman, Chandrika Mukerji
1992 Lecture Notes in Computer Science
In this paper, we present a new register allocation framework based on hierarchical cyclic interval graphs.  ...  In addition, we present a spilling algorithm that makes use of the extra information available in the interval graph representation.  ...  Note that the lifetimes of each variable are represented by a sequence of intervals, one interval for each iteration.  ...

### Random Generation and Enumeration of Proper Interval Graphs [chapter]

Toshiki Saitoh, Katsuhisa Yamanaka, Masashi Kiyomi, Ryuhei Uehara
2009 Lecture Notes in Computer Science
( ( ) ( ( ) ) ( ) ) Proper Interval Graph Unit Interval Rep. ( ( ) ( ( ) ) ( ) ) reversible This graph has only one string representation.  ...  Huang, 1996)  A connected P. I. G. has only one or two string rep.( ( ) ( ( ) ( ) ) )Proper Interval Graph Unit Interval Rep. String Rep.  Lemma 1.(X. Dell, P. Hell, J.  ...
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