If L is a language, the automaticity function A_L(n) (resp. N_L(n)) of L counts the number of states of a smallest deterministic (resp. non-deterministic) finite automaton that accepts a language that agrees with L on all inputs of length at most n. We provide bounds for the automaticity of the language of primitive words and the language of unbordered words over a k-letter alphabet. We also give a bound for the automaticity of the language of base-b representations of the irreducible […]

Section:
Automata, Logic and Semantics

The famous conjecture of Lovász and Plummer, very recently proven by Esperet et al. (2011), asserts that every cubic bridgeless graph has exponentially many perfect matchings. In this paper we improve the bound of Esperet et al. for a specific subclass of cubic bridgeless graphs called the Klee-graphs. We show that every Klee-graph with n ≥8 vertices has at least 3 *2(n+12)/60 perfect matchings.

Section:
Combinatorics

A set of vertices S is a determining set of a graph G if every automorphism of G is uniquely determined by its action on S. The determining number of G is the minimum cardinality of a determining set of G. This paper studies the determining number of Kneser graphs. First, we compute the determining number of a wide range of Kneser graphs, concretely Kn:k with n≥k(k+1) / 2+1. In the language of group theory, these computations provide exact values for the base size of the symmetric group Sn […]

Section:
Graph Theory

Karonski, Luczak, and Thomason (2004) conjecture that, for any connected graph G on at least three vertices, there exists an edge weighting from 1, 2, 3 such that adjacent vertices receive different sums of incident edge weights. Bartnicki, Grytczuk, and Niwcyk (2009) make a stronger conjecture, that each edge's weight may be chosen from an arbitrary list of size 3 rather than 1, 2, 3. We examine a variation of these conjectures, where each vertex is coloured with a sequence of edge weights. […]

Section:
Graph Theory

A biclique is a set of vertices that induce a complete bipartite graph. A graph G is biclique-Helly when its family of maximal bicliques satisfies the Helly property. If every induced subgraph of G is also biclique-Helly, then G is hereditary biclique-Helly. A graph is C4-dominated when every cycle of length 4 contains a vertex that is dominated by the vertex of the cycle that is not adjacent to it. In this paper we show that the class of hereditary biclique-Helly graphs is formed precisely by […]

Section:
Graph and Algorithms

Let g(n) denote the minimum number of edges of a maximal nontraceable (MNT) graph of order n. In 2005 Frick and Singleton (Lower bound for the size of maximal nontraceable graphs, Electronic Journal of Combinatorics, 12(1) R32, 2005) proved that g(n) = ⌈3n-22 ⌉ for n ≥54 as well as for n ∈I, where I= 12,13,22,23,30,31,38,39, 40,41,42,43,46,47,48,49,50,51 and they determined g(n) for n ≤9. We determine g(n) for 18 of the remaining 26 values of n, showing that g(n) = ⌈ 3n-22 ⌉ for n ≥54 as well as […]

Section:
Graph Theory

Let f(n,k) be the minimum number of edges that must be removed from some complete geometric graph G on n points, so that there exists a tree on k vertices that is no longer a planar subgraph of G. In this paper we show that ( 1 / 2 )n2 / k-1-n / 2≤f(n,k) ≤2 n(n-2) / k-2. For the case when k=n, we show that 2 ≤f(n,n) ≤3. For the case when k=n and G is a geometric graph on a set of points in convex position, we completely solve the problem and prove that at least three edges must be removed.

Section:
Combinatorics

Giving a planar graph G, let χ'l(G) and χ''l(G) denote the list edge chromatic number and list total chromatic number of G respectively. It is proved that if G is a planar graph without non-induced 7-cycles, then χ'l(G)≤Δ(G)+1 and χ''l(G)≤Δ(G)+2 where Δ(G)≥7.

Section:
Graph and Algorithms

A Krausz (k,m)-partition of a graph G is a decomposition of G into cliques, such that any vertex belongs to at most k cliques and any two cliques have at most m vertices in common. The m-Krausz dimension kdimm(G) of the graph G is the minimum number k such that G has a Krausz (k,m)-partition. In particular, 1-Krausz dimension or simply Krausz dimension kdim(G) is a well-known graph-theoretical parameter. In this paper we prove that the problem "kdim(G)≤3" is polynomially solvable for […]

Section:
Graph Theory

We introduce a variation of chip-firing games on connected graphs. These 'burn-off' games incorporate the loss of energy that may occur in the physical processes that classical chip-firing games have been used to model. For a graph G=(V,E), a configuration of 'chips' on its nodes is a mapping C:V→ℕ. We study the configurations that can arise in the course of iterating a burn-off game. After characterizing the 'relaxed legal' configurations for general graphs, we enumerate the 'legal' ones for […]

Section:
Graph Theory

In this paper we describe all edge-colored graphs that are fully symmetric with respect to colors and transitive on every set of edges of the same color. They correspond to fully symmetric homogeneous factorizations of complete graphs. Our description completes the work done in our previous paper, where we have shown, in particular, that there are no such graphs with more than 5 colors. Using some recent results, with a help of computer, we settle all the cases that was left open in the previous […]

Section:
Graph Theory

We give a linear-time algorithm that checks for isomorphism between two 0-1 matrices that obey the circular-ones property. Our algorithm is similar to the isomorphism algorithm for interval graphs of Lueker and Booth, but works on PC trees, which are unrooted and have a cyclic nature, rather than with PQ trees, which are rooted. This algorithm leads to linear-time isomorphism algorithms for related graph classes, including Helly circular-arc graphs, Γ circular-arc graphs, proper circular-arc […]

Section:
Discrete Algorithms

A b-coloring of a graph G by k colors is a proper vertex coloring such that each color class contains a color-dominating vertex, that is, a vertex having neighbors in all other k-1 color classes. The b-chromatic number χb(G) is the maximum integer k for which G has a b-coloring by k colors. Let Cnr be the rth power of a cycle of order n. In 2003, Effantin and Kheddouci established the b-chromatic number χb(Cnr) for all values of n and r, except for 2r+3≤n≤3r. For the missing cases they presented […]

Section:
Graph Theory

Following the track of polyominoes, in particular the column-by-column construction of Temperley and its interpretation in terms of functional equations due to Bousquet-Mélou, we introduce a generic method for the enumeration of classes of directed polycubes the strata of which satisfy some property P. This method is applied to the enumeration of two new families of polycubes, the s-directed polycubes and the vertically-convex s-directed polycubes, with respect to width and volume. The case of […]

Section:
Combinatorics