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### P3-isomorphisms for graphs

R. E. L. Aldred, M. N. Ellingham, R. L. Hemminger, P. Jipsen
1997 Journal of Graph Theory
P 3 -ISOMORPHISMS FOR GRAPHS 37 THORNS, DIAMONDS, SWAPS AND DISCONNECTED GRAPHS In this section we establish much of the necessary terminology for our work.  ...  We also characterize all isomorphisms between P 3 -graphs in terms of the original graphs.  ...  In Section 2, we discuss some basic constructions for noninduced P 3 -isomorphisms and the situation for disconnected graphs.  ...

### The generation of minimal triangle graphs

Rufus Bowen
1967 Mathematics of Computation
T is a minimal triangle set (MTS) of order n if (i) GiT) has n vertices and is connected, and (ii) G(T -\t) ) satisfies (i) for no t in T.  ...  In such a case G(T) is a minimal triangle graph (MTG) of order n. Our problem is to generate all (nonisomorphic) MTS's of a given order or equivalently (by Theorem 1) all  ...  We are left with the problem of sorting out isomorphic MTS's. For this purpose a routine was developed for testing two graphs for isomorphism (described in the next paragraph ).  ...

### The Generation of Minimal Triangle Graphs

Robert Bowen
1967 Mathematics of Computation
T is a minimal triangle set (MTS) of order n if (i) GiT) has n vertices and is connected, and (ii) G(T -\t) ) satisfies (i) for no t in T.  ...  In such a case G(T) is a minimal triangle graph (MTG) of order n. Our problem is to generate all (nonisomorphic) MTS's of a given order or equivalently (by Theorem 1) all  ...  We are left with the problem of sorting out isomorphic MTS's. For this purpose a routine was developed for testing two graphs for isomorphism (described in the next paragraph ).  ...

### Randomly decomposable graphs

Sergio Ruiz
1985 Discrete Mathematics
Ruiz and a rational number a, such that 0 < a ~< 1, does there exist a regular Hdecomposable graph G for which i(G: H)= ~?  ...  A graph G is randomly P3-decomposable if and only if each component of G is isomorphic to C4 or a star of even size. Proof.  ...  If G is eulerian, then every two consecutive edges of an eulerian circuit account for a copy of P3, producing the desired P3-decomposition.  ...

### The distance between a graph and its complement

1987 Czechoslovak Mathematical Journal
If n ^ 4, then P" contains an induced subgraph isomorphic to P3; the subgraph induced by the same vertex set in P" is also isomorphic to P3. The graph P3 has four vertices and thus ô{P") g n -4.  ...  The assertions for Pj and P2 are evident. The path P3 is a self-complementary graph.  ...

### Page 12 of Mathematical Reviews Vol. , Issue 2004b [page]

2004 Mathematical Reviews
The results include: a characterization of directed graphs D with P3(D) isomorphic to D; a characterization of directed graphs D with P3(D) Eulerian or Hamiltonian; and that an isomorphism between P3(D  ...  ,) and P3(D>) almost always implies an isomorphism between D, and D>.  ...

### Stability of cartesian products

Julie Sims, D.A Holton
1978 Journal of combinatorial theory. Series B (Print)
Further we show that except for certain (listed) restricted graphs, comected cartesian products are semi-stable at every vertex.  ...  Finally, we show that the cartesian product of any two graphs is not semi-stable if and only if one of them is totally disconnected and the other is not semi-stable. 258  ...  But this graph is isomorphic to M i.e. M g P2 x KT for some subgraph M of H. Also, by P7, So in case (a), either we have a contradiction, or w1 is a perturbation point of H.  ...

### Generations of triangulations of the sphere

Rufus Bowen, Stephen Fisk
1967 Mathematics of Computation
First we develop a method for generating all triangulations with JV vertices which may yield several triangulations of the same isomorphism type, and then wre describe an isomorphism routine for eliminating  ...  The slight complication here is needed to insure that T' is a triangulation; for if Pi were adjacent to P3 in T, then T' would have multiple edges and would not be a triangulation.  ...  The authors are indebted to the University of California (Berkeley) Computer Center for the use of its machines. University of California Berkeley, California 94533  ...

### Generation of Triangulations of the Sphere

Robert Bowen, Stephen Fisk
1967 Mathematics of Computation
First we develop a method for generating all triangulations with JV vertices which may yield several triangulations of the same isomorphism type, and then wre describe an isomorphism routine for eliminating  ...  The slight complication here is needed to insure that T' is a triangulation; for if Pi were adjacent to P3 in T, then T' would have multiple edges and would not be a triangulation.  ...  The authors are indebted to the University of California (Berkeley) Computer Center for the use of its machines. University of California Berkeley, California 94533  ...

### Page 2095 of Mathematical Reviews Vol. , Issue 98D [page]

1998 Mathematical Reviews
They also characterize all isomorphisms between P3-graphs in terms of the original graphs.  ...  In the present paper the authors show that connected finite simple graphs G and H with isomorphic P3-graphs are either isomorphic or part of three exceptional families.  ...

### Some perfect coloring properties of graphs

Claude A Christen, Stanley M Selkow
1979 Journal of combinatorial theory. Series B (Print)
For any graph G, the following statements are equivalent: (5) G is *p-perfect. (6) G is z,bx-perfect. (7) G does not contain an induced subgraph isomorphic to one of the graphs P4 , P3 v P2 or Pz V P2  ...  By inspection, one can verify that in any case a subset of these points induce a graph isomorphic to P4 , P3 u P, . or P, u Pz u Pz .  ...

### ON SMALLEST ORDER OF TRIANGLE-FREE GRAPHS WITH PRESCRIBED (3, k)-DEFECTIVE CHROMATIC NUMBER

Mudin Simanihuruk
2018 Far East Journal of Mathematical Sciences (FJMS)
We show that f(3, 1) = 9 and characterize the corresponding minimal graphs. For m ~ 4, we present lower and upper bounds for f(m,l).  ...  The k-defective chromatic number Xk(G) of a graph G is the least positive integer m for which G is (m,k)-colourable. Let f(m,k) be the smallest order of a triangle-free graph G such that X k (G) = m.  ...  For n I and 2 the graphs Kl and P3, respectively, have the required properties. Assume that R is a triangle-free graph of order p with Xl (R) = k, where k :?: 3.  ...

### Page 2273 of Mathematical Reviews Vol. , Issue 85f [page]

1985 Mathematical Reviews
A graph is vertex pancyclic if each of its vertices lies on a cycle of length |, for each | with 3 <1< |V(G)|. Y is a graph isomorphic to S(K;,3) minus an end vertex.)  ...  Enomoto and the author [same journal 20 (1981), 117-141; MR 83a:05077], namely (1.1) |E(G)| > jn — 42 holds for a P3- connected graph G with n vertices.  ...

### Page 3936 of Mathematical Reviews Vol. , Issue 96g [page]

1996 Mathematical Reviews
Summary: “For graphs G and G’ with minimum degree at least 3 and satisfying one of three other conditions, we prove that any isomorphism from the P3-graph P3(G) onto P3(G’) can be induced by a (vertex-  ...  ) isomorphism of G onto G’.  ...

### The structure of the models of decidable monadic theories of graphs

D. Seese
1991 Annals of Pure and Applied Logic
., The structure of the models of decidable monadic theories of graphs, Annals of Pure and Applied Logic 53 (1991) 169-195.  ...  Let f be an isomorphism from Q2,+1 to GA and define &:= {f(e): e E &} for i = 1, 2, 3. Now assume that PI,i, P2,i and P3,i are already defined for an i with 0 < i < 1.  ...  There was also conjectured a similar results for planar graphs with no minor isomorphic to Q, (for n 2 3). This results holds also for infinite' graphs G. Theorem 2 (Thomas [70, 73] ).  ...
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