On Light Graphs in 3-Connected Plane Graphs Without Triangular or Quadrangular Faces

 We prove that each 3-connected plane graph G without triangular or quadrangular faces either contains a k-path P _k , a path on k vertices, such that each of its k vertices has degree ≤5/3k in G or does not contain any k-path. We also prove that each 3-connected pentagonal plane graph G which has a k-cycle, a cycle on k vertices, k∈ {5,8,11,14}, contains a k-cycle such that all its vertices have, in G, bounded degrees. Moreover, for all integers k and m, k≥ 3, k∉ {5,8,11,14} and m≥ 3, we present a graph in which every k-cycle contains a vertex of degree at least m. Received: June 29, 1998 Final version received: April 11, 2000     

Hamilton ?-cycles in uniform hypergraphs

We say that a k-uniform hypergraph C is an ?-cycle if there exists a cyclic ordering of the vertices of C such that every edge of C consists of k consecutive vertices and such that every pair of consecutive edges (in the natural ordering of the edges) intersects in precisely ? vertices. We prove that if 1??<k and k−? does not divide k then any k-uniform hypergraph on n vertices with minimum degree at least contains a Hamilton ?-cycle. This confirms a conjecture of Hàn and Schacht. Together with results of Rödl, Ruciński and Szemerédi, our result asymptotically determines the minimum degree which forces an ?-cycle for any ? with 1??<k.     

A bound on the generalized competition index of a primitive matrix using Boolean rank

For a positive integer m where 1?m?n, the m-competition index (generalized competition index) of a primitive digraph is the smallest positive integer k such that for every pair of vertices x and y, there exist m distinct vertices v₁,v₂,…,v_m such that there are directed walks of length k from x to v_i and from y to v_i for 1?i?m. The m-competition index is a generalization of the scrambling index and the exponent of a primitive digraph. In this study, we determine an upper bound on the m-competition index of a primitive digraph using Boolean rank and give examples of primitive Boolean matrices that attain the bound.     

b-colouring outerplanar graphs with large girth

A b-colouring of a graph is a colouring of its vertices such that every colour class contains a vertex that has a neighbour in all other classes. The b-chromatic number of a graph is the largest integer $k$ such that the graph has a $b$-colouring with $k$ colours. We show here how to compute in polynomial time the $b$-chromatic number of an outerplanar graph of girth at least $8$. This generalizes the seminal result of Irving and Manlove on trees.     

On realizations of point determining graphs, and obstructions to full homomorphisms

A graph is point determining if distinct vertices have distinct neighbourhoods. A realization of a point determining graph H is a point determining graph G such that each vertex-removed subgraph G-x which is point determining, is isomorphic to H. We study the fine structure of point determining graphs, and conclude that every point determining graph has at most two realizations.A full homomorphism of a graph G to a graph H is a vertex mapping f such that for distinct vertices u and v of G, we have uv an edge of G if and only if f(u)f(v) is an edge of H. For a fixed graph H, a full H-colouring of G is a full homomorphism of G to H. A minimal H-obstruction is a graph G which does not admit a full H-colouring, such that each proper induced subgraph of G admits a full H-colouring. We analyse minimal H-obstructions using our results on point determining graphs. We connect the two problems by proving that if H has k vertices, then a graph with k+1 vertices is a minimal H-obstruction if and only if it is a realization of H. We conclude that every minimal H-obstruction has at most k+1 vertices, and there are at most two minimal H-obstructions with k+1 vertices.We also consider full homomorphisms to graphs H in which loops are allowed. If H has ? loops and k vertices without loops, then every minimal H-obstruction has at most (k+1)(?+1) vertices, and, when both k and ? are positive, there is at most one minimal H-obstruction with (k+1)(?+1) vertices.In particular, this yields a finite forbidden subgraph characterization of full H-colourability, for any graph H with loops allowed.     

Independent sets ink-chromatic graphs

A graph is said to have propertyP _k if in eachk-colouring ofG using allk colours there arek independent vertices having all colours. An (unpublished) suggestion of P. Erdős is answered in the affirmative: For eachk≧3 there is a k-critical graph withP _k . With the aid of a construction of T. Gallaik-chromatic graphs (k≧7) withP _k orP _k+1 of arbitrarily high connectivity are obtained. The main result is: Eachk-chromatic graph (k≧3) of girth ≧6 hasP _k or is a circuit of length 7. Dedicated to Paul Erdős on his seventieth birthday     

Enclosings of λ-fold 4-cycle systems

In this paper we solve the problem of enclosing a λ-fold 4-cycle system of order v into a (λ + m)-fold 4-cycle system of order v + u for all m > 0 and u ≥ 1. An ingredient is constructed that is of interest on its own right, namely the problem of finding equitable partial 4-cycle systems of λ K _v . This supplementary solution builds on a result of Raines and Staniszlo.     

On list critical graphs

In this paper we discuss some basic properties of k-list critical graphs. A graph G is k-list critical if there exists a list assignment L for G with |L(v)|=k−1 for all vertices v of G such that every proper subgraph of G is L-colorable, but G itself is not L-colorable. This generalizes the usual definition of a k-chromatic critical graph, where L(v)={1,…,k−1} for all vertices v of G. While the investigation of k-critical graphs is a well established part of coloring theory, not much is known about k-list critical graphs. Several unexpected phenomena occur, for instance a k-list critical graph may contain another one as a proper induced subgraph, with the same value of k. We also show that, for all 2≤p≤k, there is a minimal k-list critical graph with chromatic number p. Furthermore, we discuss the question, for which values of k and n is the complete graph K_nk-list critical. While this is the case for all 5≤k≤n, K_n is not 4-list critical if n is large.     

Critical hypergraphs for the weak chromatic number

Instead of removing a vertex or an edge from a hypergraph H, one may add to some edges of H new vertices (not necessarily belonging to V_H). The weak chromatic number of H tends to drop by this operation. This suggests the definition of an order relation ≥ on the set $\text{S}$ of all Sperner hypergraphs on a universal set V of vertices. The corresponding criticality study leads to unifying and interesting results: reconstruction of critical hypergraphs and two general characterizations of k-chromatic critical hypergraphs (k ≥ 3), from which a special characterization of 3-chromatic critical hypergraphs can be derived.     

Local condition for planar graphs of maximum degree 7 to be 8-totally colorable

The total chromatic number of a graph G, denoted by χ^″(G), is the minimum number of colors needed to color the vertices and edges of G such that no two adjacent or incident elements get the same color. It is known that if a planar graph G has maximum degree Δ≥9, then χ^″(G)=Δ+1. In this paper, we prove that if G is a planar graph with maximum degree 7, and for every vertex v, there is an integer k_v∈{3,4,5,6} so that v is not incident with any k_v-cycle, then χ^″(G)=8.     

ON UNIQUELY -G k-COLOURABLE GRAPHS

Abstract

Given graphs F and G and a nonnegative integer k, a map Π: V(F) → + {lm …, k} is a -G k-colouring of F if the subgraphs induced by each colour class do not contain G as an induced subgraph; F is -G k-chromatic if F has a -G k-colouring but no -G (k—1)-colouring. Further, we say F is uniquely -G k-colourable if and only if F is -G k-chromatic and, up to a permutation of colours, it has only one -G k-colouring. Such notions are extensions of the well known corresponding definitions from chromatic theory. In a previous paper (J. Graph. Th. 11 (1987), 87–99), the authors conjectured that for all graphs G of order at least two and all nonnegative integers k there exist uniquely -G k-colourable graphs. We show here that the conjecture holds whenever G or its complement is 2-connected.     

Generalized competition index of a primitive digraph

For positive integers k and m, and a digraph D, the k-step m-competition graph of D has the same set of vertices as D and an edge between vertices x and y if and only if there are distinct m vertices v₁,v₂,…,v_m in D such that there are directed walks of length k from x to v_i and from y to v_i for 1?i?m. In this paper, we present the definition of m-competition index for a primitive digraph. The m-competition index of a primitive digraph D is the smallest positive integer k such that is a complete graph. We study m-competition indices of primitive digraphs and provide an upper bound for the m-competition index of a primitive digraph.     

Equipartite gregarious 6- and 8-cycle systems

A k-cycle decomposition of a complete multipartite graph is said to be gregarious if each k-cycle in the decomposition has its vertices in k different partite sets. Equipartite gregarious 3-cycle systems are 3-GDDs, and necessary and sufficient conditions for their existence are known (see for instance the CRC Handbook of Combinatorial Designs, 1996, C.J. Colbourn, J.H. Dinitz (Eds.), Section III 1.3). The cases of equipartite and of almost equipartite 4-cycle systems were recently dealt with by Billington and Hoffman. Here, for both 6-cycles and for 8-cycles, we give necessary and sufficient conditions for existence of a gregarious cycle decomposition of the complete equipartite graph K_n(a) (with n parts, n?6 or n?8, of size a).     

Adaptable choosability of planar graphs with sparse short cycles

Given a (possibly improper) edge colouring F of a graph G, a vertex colouring of G is adapted toF if no colour appears at the same time on an edge and on its two endpoints. A graph G is called (for some positive integer k) if for any list assignment L to the vertices of G, with |L(v)|≥k for all v, and any edge colouring F of G, G admits a colouring c adapted to F where c(v)∈L(v) for all v. This paper proves that a planar graph G is adaptably 3-choosable if any two triangles in G have distance at least 2 and no triangle is adjacent to a 4-cycle.     

A new class for large sets of almost Hamilton cycle decompositions

A k-cycle system of order v with index λ, denoted by CS(v, k, λ), is a collection A of k-cycles (blocks) of K _v such that each edge in K _v appears in exactly λ blocks of A. A large set of CS(v, k, λ)s is a partition of the set of all k-cycles of K _v into CS(v, k, λ)_s, and is denoted by LCS(v, k, λ). A (v ?1)-cycle in K _v is called almost Hamilton. The completion of the existence problem for LCS(v, v?1, λ) depends only on one case: all v ≥ 4 for λ = 2. In this paper, it is shown that there exists an LCS(v, v ? 1, 2) for all v ≡ 2 (mod 4), v ≥ 6.     

Linear layouts measuring neighbourhoods in graphs

In this paper we introduce the graph layout parameter neighbourhood-width as a variation of the well-known cut-width. The cut-width of a graph G=(V,E) is the smallest integer k, such that there is a linear layout ?:V→{1,…,|V|}, such that for every 1?i<|V| there are at most k edges {u,v} with ?(u)?i and ?(v)>i. The neighbourhood-width of a graph is the smallest integer k, such that there is a linear layout ?, such that for every 1?i<|V| the vertices u with ?(u)?i can be divided into at most k subsets each members having the same neighbourhood with respect to the vertices v with ?(v)>i.We show that the neighbourhood-width of a graph differs from its linear clique-width or linear NLC-width at most by one. This relation is used to show that the minimization problem for neighbourhood-width is NP-complete.Furthermore, we prove that simple modifications of neighbourhood-width imply equivalent layout characterizations for linear clique-width and linear NLC-width.We also show that every graph of path-width k or cut-width k has neighbourhood-width at most k+2 and we give several conditions such that graphs of bounded neighbourhood-width have bounded path-width or bounded cut-width.     

Geodesic-pancyclic graphs

A shortest path connecting two vertices u and v is called a u-v geodesic. The distance between u and v in a graph G, denoted by d_G(u,v), is the number of edges in a u-v geodesic. A graph G with n vertices is panconnected if, for each pair of vertices u,v∈V(G) and for each integer k with d_G(u,v)?k?n-1, there is a path of length k in G that connects u and v. A graph G with n vertices is geodesic-pancyclic if, for each pair of vertices u,v∈V(G), every u-v geodesic lies on every cycle of length k satisfying max{2d_G(u,v),3}?k?n. In this paper, we study sufficient conditions of geodesic-pancyclic graphs. In particular, we show that most of the known sufficient conditions of panconnected graphs can be applied to geodesic-pancyclic graphs.     

Interpolation theorem for the number of pendant vertices of connected spanning subgraphs of equal size

At the 4th International Graph Theory Conference 1980, G. Chartrand posed the following problem: If a (connected) graph G contains spanning trees with m and n pendant vertices, respectively, with m < n, does G contain a spanning tree with k pendant vertices for every integer k, where m < k < n? Recently, S. Schuster showed that the answer is yes. Several variations of this interpolation theorem will be given including the following generalization: If a connected graph G contains connected spanning subgraphs of size r with m and n pendant vertices, respectively, with m < n, then G contains a connected spanning subgraph of size r with k pendant vertices for every integer k, where m < k < n.     

Graph powers and k-ordered Hamiltonicity

It is known that if G is a connected simple graph, then G³ is Hamiltonian (in fact, Hamilton-connected). A simple graph is k-ordered Hamiltonian if for any sequence v₁, v₂,…,v_k of k vertices there is a Hamiltonian cycle containing these vertices in the given order. In this paper, we prove that if k?4, then G^⌊3k/2⌋-2 is k-ordered Hamiltonian for every connected graph G on at least k vertices. By considering the case of the path graph P_n, we show that this result is sharp. We also give a lower bound on the power of the cycle C_n that guarantees k-ordered Hamiltonicity.